TREATISE 


ON 


HYDRAULICS 


BY 

MANSFIELD    MERRIMAN 

PROFESSOR  OF  CIVIL  ENGINEERING  IN  LEHIGH  UNIVERSITY 


EIGHTH  EDITION,  REWRITTEN  AND  ENLARGED 

NINTH    THOUSAND 
TOTAL    ISSUE,    TWENTY-TWO    THOUSAND 


NEW  YORK 

JOHN   WILEY  &   SONS 

LONDON:   CHAPMAN    &    HALL,  LIMITED 
1905 


COPYRIGHT,  1903, 

BY 
MANSFIELD  MERRIMAN 

First  Edition,  February,  1889 
Second  Edition,  March,  1890 
Third  Edition,  December,  1890 

Fourth  Edition,  July,  1891  ;  reprinted  1893  (twice),  1894  (twice) 
Fifth  Edition,  enlarged,  March,  1895  ;  reprinted  1895,  1896,  1897,  18 

Sixth  Edition,  August,  1898  ;  reprinted  1899 
Seventh  Edition,  September,  1900 ;  reprinted  1901,  1902  (twice) 

Eighth  Edition,  rewritten  and  enlarged,  May,  1903 
Reprinted  December,  1903  ;  October,  1904 ;  September,  1905 

ALL  RIGHTS  RESERVED 


ROBERT  DRUMMOND,    PRINTER,  NEW  YORK. 


PREFACE 

Since  the  publication  of  the  first  edition  of  this  treatise,  in 
1889,  many  advances  have  been  made  in  Hydraulics.  Some  of 
these  have  been  briefly 'noted  in  later  editions,  but  to  properly 
record  and  correlate  them  it  has  now  become  necessary  to 
rewrite  and  reset  the  book.  In  so  doing  the  author  has  en- 
deavored to  incorporate  other  features  that  have  been  suggested 
to  him  by  teachers  and  engineers,  to  whom  he  here  expresses 
his  thanks.  All  of  these  suggestions  could  not  be  followed,  foi 
thereby  the  work  would  have  been  expanded  to  two  volumes. 
Indeed  the  question  as  to  what  should  be  left  out  has  often  been 
a  more  difficult  one  than  that  as  to  what  should  be  inserted,  and 
the  author  has  made  the  decision  from  the  point  of  view  of  the 
probable  benefit  that  may  accrue  to  students  in  engineering 
colleges  and  to  engineers  in  ordinary  conditions  of  practice. 

The  same  plan  of  arrangement  as  in  former  editions  has 
been  followed,  but  two  new  chapters  have  been  added,  one  on 
Hydraulic  Instruments  and  Observations  which  treats  of  the 
methods  of  measuring  pressures  and  velocities,  and  another  on 
Pumps  and  Pumping  in  which  the  various  machines  for  raising 
water  are  discussed  from  a  hydraulic  point  of  view.  Among 
the  new  topics  introduced  in  the  other  chapters  may  be  noted 
the  vortex  whirl  that  occurs  in  emptying  a  vessel,  new  coeffi- 
cients for  dams  and  for  steel  and  wood  pipes,  the  loss  of  head  in 
pipes  due  to  curvature,  branched  circuits  or  diversions  in  pipe 
systems,  the  influence  of  piers  in  producing  backwater,  canals 
for  water-power  plants,  discharge  curves  for  rivers,  the  tidal 
and  the  land  bore,  w^ter-supply  estimates,  water  hammer  in 
pipes,  the  stability  of  a  ship,  and  hydraulic-electric  analogies. 
Many  new  examples  and  problems  are  given  and  in  these  the 
author  has  endeavored  not  only  to  exemplify  the  theory  of  the 
subject,  but  also  to  illustrate  the  conditions  of  actual  practice. 
Historical  notes  and  references  to  hydraulic  literature  are 
presented  with  greater  fullness  than  before.  The  number  of 

iii 


iv  PREFACE 

articles  has  been  increased  from  164  to  196,  the  number  of  cuts 
from  123  to  199,  the  number  of  tables  from  29  to  55,  the  num- 
ber of  problems  from  190  to  330,  and  the  number  of  pages  from 
x  +  427  to  viii +585. 

Many  letters  from  foreign  countries  have  urged  the  author 
to  introduce  the  metric  system  of  measures  into  the  book.  To 
meet  this  demand,  the  most  important  metric  data,  coefficients, 
formulas,  and  problems  are  given  at  the  end  of  each  chapter, 
and  the  student  who  follows  these  will  have  no  occasion  to 
transform  English  measures,  but  may  learn  to  think  in  metric 
units  and  to  use  them  without  hesitation.  The  hydraulic  tables 
have  been  gathered  together  at  the  end  of  the  volume,  instead 
of  being  scattered  throughout  the  text,  and  it  is  believed  that 
this  plan  affords  the  advantage  that  a  table  can  be  more  readily 
found.  The  most  important  tables  are  presented  both  in  the 
English  and  in  the  metric  system,  the  latter  not  being  a  mere 
transformation  of  the  former,  but  being  arranged  to  be  used 
with  metric  arguments. 

In  former  editions  of  this  work,  as  in  most  other  books,  the 
numbers  of  the  articles,  formulas,  cuts,  and  problems  were  con- 
secutive and  independent.  In  this  edition,  however,  only  the 
articles  are  numbered  consecutively,  while  the  number  of  any 
formula,  cut,  or  problem  agrees  with  that  of  the  article  and  this 
number  is  placed  at  the  top  of  the  right-hand  page.  While  the 
main  purpose  in  rewriting  the  book  has  been  to  keep  it  abreast 
with  modern  progress,  the  attempt  has  also  been  made  to  pre- 
sent the  subject  more  concisely  and  clearly  than  before,  in 
order  to  advance  the  interests  of  thorough  education  and  to 
promote  sound  engineering  practice. 

MANSFIELD  MERRIMAN. 

UNIVERSITY,  SOUTH  BETHI.EHEM,  PA., 
April,  1903. 


CONTENTS 

CHAPTER  I.     FUNDAMENTAL  DATA 

PAGES 

ART.  1.  HISTORICAL  NOTES.  2  UNITS  OF  MEASURE.  3.  PHYSICAL 
PROPERTIES  OF  WATER.  4.  THE  WEIGHT  OF  WATER.  5.  ATMOSPHERIC 
PRESSURE.  6.  COMPRESSIBILITY  OF  WATER.  7.  ACCELERATION  DUB 
TO  GRAVITY.  8.  NUMERICAL  COMPUTATIONS.  9.  DATA  IN  THE  METRIC 
SYSTEM 1-21 

CHAPTER  II.     HYDROSTATICS 

ART.  10.  TRANSMISSION  OF  PRESSURE.  11.  HEAD  AND  PRESSURE. 
12.  Loss  OF  WEIGHT  IN  WATER.  13.  DEPTH  OF  FLOTATION.  14.  STA- 
BILITY OF  FLOTATION.  15.  NORMAL  PRESSURE.  16.  PRESSURE  IN  A 
GIVEN  DIRECTION.  17.  CENTER  OF  PRESSURE  ON  RECTANGLES.  18. 
GENERAL  RULE  FOR  CENTER  OF  PRESSURE.  19.  PRESSURE  ON  GATES 
AND  DAMS.  20.  HYDROSTATICS  IN  METRIC  MEASURES 22-45 

CHAPTER  III.     THEORETICAL  HYDRAULICS 

ART.  21.  LAWS  OF  FALLING  BODIES.  22.  VELOCITY  OF  FLOW  FROM 
ORIFICES.  23.  DISCHARGE  FROM  SMALL  ORIFICES.  24.  FLOW  UNDER 
PRESSURE.  25.  INFLUENCE  OF  VELOCITY  OF  APPROACH.  26.  EMPTY- 
ING A  VESSEL.  27.  THE  PATH  OF  A  JET.  28.  THE  ENERGY  OF  A  JET. 
29.  IMPULSE  AND  REACTION  OF  A  JET.  30.  ABSOLUTE  AND  RELATIVE 
VELOCITIES.  31.  FLOW  FROM  A  REVOLVING  VESSEL.  32.  STEADY 
FLOW  IN  SMOOTH  PIPES.  33.  COMPUTATIONS  IN  METRIC  MEASURES.  46-80 

CHAPTER  IV.      INSTRUMENTS  AND  OBSERVATIONS 

ART.  34.  GENERAL  CONSIDERATIONS.  35.  THE  HOOK  GAGE.  36. 
PRESSURE  GAGES.  37.  DIFFERENTIAL  PRESSURE  GAGES.  38.  WATER 
METERS.  39.  MEASUREMENT  OF  VELOCITY  40  THE  CURRENT  METER. 
41.  THE  PITOT  TUBE.  42.  DISCUSSION  OF  OBSERVATIONS.  .  .  .  81-108 

CHAPTER  V.     FLOW  THROUGH  ORIFICES 

ART.  43.  STANDARD  ORIFICES.  44.  COEFFICIENT  OF  CONTRACTION. 
45.  COEFFICIENT  OF  VELOCITY.  46.  COEFFICIENT  OF  DISCHARGE. 
47.  CIRCULAR  VERTICAL  ORIFICES.  48.  SQUARE  VERTICAL  ORIFICES. 


vi  CONTENTS 


PAGI 


49.  RECTANGULAR  VERTICAL  ORIFICES.  50.  THE  MINER'S  INCH. 
51.  VELOCITY  OF  APPROACH.  52.  SUBMERGED  ORIFICES.  53.  SUP- 
PRESSION OF  THE  CONTRACTION.  54.  ORIFICES  WITH  ROUNDED  EDGES. 
55.  WATER  MEASUREMENT  BY  ORIFICES.  56.  Loss  OF  ENERGY  OR 
HEAD.  •  57.  DISCHARGE  UNDER  A  DROPPING  HEAD.  58.  EMPTYING 
AND  FILLING  A  CANAL  LOCK.  59.  METRIC  COMPUTATIONS.  .  .  .  109-140 

CHAPTER  VI.     FLOW  OVER  WEIRS 

ART.  60.  DESCRIPTION  OF  WEIRS.  61.  FORMULAS  FOR  DISCHARGE. 
62.  VELOCITY  OF  APPROACH.  63.  WEIRS  WITH  END  CONTRACTIONS. 
64.  WEIRS  WITHOUT  END  CONTRACTIONS.  65.  FRANCIS'  FORMULAS. 
66.  SUBMERGED  WEIRS.  67.  ROUNDED  AND  WIDE  CRESTS.  68.  WASTE 
WEIRS  AND  DAMS.  69.  THE  SURFACE  CURVE.  70.  TRIANGULAR 
WEIRS.  71.  TRAPEZOIDAL  WEIRS.  72.  COMPUTATIONS  IN  THE  METRIC 
SYSTEM 141-169 

CHAPTER  VII.     FLOW  THROUGH  TUBES 

ART.  73.  Loss  OF  ENERGY  OR  HEAD.  74.  Loss  DUE  TO  EXPANSION 
OF  SECTION.  75.  Loss  DUE  TO  CONTRACTION  OF  SECTION.  76.  THE 
STANDARD  SHORT  TUBE.  77.  CONICAL  CONVERVING  TUBES.  78  IN- 
WARD PROJECTING  TUBES.  79.  DIVERGING  AND  COMPOUND  TUBES. 
80.  NOZZLES  AND  JETS  81.  LOST  HEAD  IN  LONG  TUBES.  -82.  IN- 
CLINED TUBES  AND  PIPES.  83.  VELOCITIES  IN  A  CROSS-SECTION. 
84.  COMPUTATIONS  IN  METRIC  MEASURES 170-203 

CHAPTER  VIII.     FLOW  THROUGH  PIPES 

ART.  85.  FUNDAMENTAL  IDEAS.  86.  Loss  OF  HEAD  IN  FRICTION. 
87.  Loss  OF  HEAD  IN  CURVATURE.  88.  OTHER  LOSSES  OF  HEAD. 
89.  FORMULA  FOR  MEAN  VELOCITY.  90.  COMPUTATION  OF  DISCHARGE. 
91.  COMPUTATION  OF  DIAMETER.  92.  SHORT  PIPES.  93.  LONG  PIPES. 
91.  PIEZOMETER  MEASUREMENTS.  95.  THE  HYDRAULIC  GRADIENT. 
93.  A  COMPOUND  PIPE.  97.  A  PIPE  WITH  NOZZLE.  98.  HOUSE  SER- 
VICE PIPES.  99:  WATER  MAINS  IN  TOWNS.  100.  BRANCHES  AND 
DIVERSIONS.  101.  RIVETED  AND  WOOD  PIPES.  102  FIRE  HOSE. 
103.  OTHER  FORMULAS  FOR  FLOW  IN  PIPES.  104.  COMPUTATIONS  IN 
METRIC  MEASURES 204-264 

CHAPTER  IX.     FLOW  IN  CONDUITS 

ART  105.  DEFINITIONS.  106.  FORMULA  FOR  MEAN  VELOCITY. 
107  CIRCULAR  CONDUITS,  FULL  OR  HALF-FULL.  108.  CIRCULAR  CON- 
DUITS, PARTLY  FULL.  109.  RECTANGULAR  SECTIONS.  110.  TRAPEZOI- 
DAL SECTIONS.  111.  KUTTER'S  FORMULA.  112.  SEWERS.  113. 
DITCHES  AND  CANALS.  114.  LARGE  STEEL  AND  WOOD  PIPES.  115. 
BAZIN'S  FORMULA.  116.  OTHER  FORMULAS  FOR  CONDUITS.  117. 
LOSSES  OF  HEAD.  118.  VELOCITIES  IN  A  CROSS-SECTION.  119.  COM- 
PUTATIONS IN  METRIC  MEASURES.  .  265-304 


CONTENTS  vii 

CHAPTER  X.     THE  FLOW  OF  RIVERS 

PAGBS 

ART.  120.  GENERAL  CONSIDERATIONS.  121.  VELOCITIES  IN  A 
CROSS-SECTION.  122.  VELOCITY  MEASUREMENTS.  123.  GAGING  THE 
DISCHARGE.  124.  APPROXIMATE  GAGINGS.  125.  COMPARISON  OF 
METHODS.  126.  VARIATIONS  IN  DISCHARGE.  127.  TRANSPORTING 
CAPACITY  OF  CURRENTS.  128.  INFLUENCE  OF  DAMS  AND  PIERS. 
129.  NON-UNIFORM  STEADY  FLOW.  130.  THE  SURFACE  CURVE,  131. 
THE  BACKWATER  CURVE.  132.  THE  DROP-DOWN  CURVE.  133.  THE 
JUMP  AND  THE  BORE 305-346 

CHAPTER  XI.     WATER  SUPPLY  AND  WATER  POWER 

ART.  134.  RAINFALL  AND  EVAPORATION.  135.  GROUND  WATER 
AND  RUNOFF.  136.  ESTIMATES  FOR  WATER  SUPPLY.  137.  ESTI- 
MATES FOR  WATER  POWER.  138.  WATER  DELIVERED  TO  A  MOTOR. 
139.  EFFECTIVE  HEAD  ON  A  WHEEL.  140.  MEASUREMENT  OF  EFFEC- 
TIVE POWER.  141.  TESTS  OF  TURBINE  WHEELS.  142.  FACTS  CON- 
CERNING WATER  POWER 347-374 

CHAPTER  XII.     DYNAMIC  PRESSURE  OF  WATER 

ART.  143.  DEFINITIONS  AND  PRINCIPLES.  144.  EXPERIMENTS  ON 
IMPULSE  AND  REACTION.  145.  SURFACES  AT  REST.  146.  IM- 
MERSED BODIES.  147.  CURVED  PIPES  AND  CHANNELS.  148.  WATER 
HAMMER  IN  PIPES.  149  MOVING  VANES.  150.  WORK  DERIVED 
FROM  MOVING  VANES.  151.  REVOLVING  VANES.  152  WORK  DE- 
RIVED FROM  REVOLVING  VANES.  153.  REVOLVING  TUBES.  .  .  375-411 

CHAPTER  XIII.     WATER  WHEELS 

ART.  154.  CONDITIONS  OF  HIGH  EFFICIENCY.  155.  OVERSHOT 
WHEELS.  156.  BREAST  WHEELS.  157.  UNDERSHOT  WHEELS.  158. 
VERTICAL  IMPULSE  WHEELS.  159.  HORIZONTAL  IMPULSE  WHEELS. 
160.  DOWNWARD-FLOW  IMPULSE  WHEELS.  161.  NOZZLES  FOR  IM- 
PULSE WHEELS.  162.  SPECIAL  FORMS  OF  WHEELS 412-436 

CHAPTER  XIV.     TURBINES 

ART.  163.  THE  REACTION  WHEEL.  164.  CLASSIFICATION  OF  TUR- 
BINES. 165.  REACTION  TURBINES.  166.  FLOW  THROUGH  REACTION 
TURBINES.  167.  THEORY  OF  REACTION  TURBINES.  168.  DESIGN  OF 
REACTION  TURBINES.  169.  GUIDES  AND  VANES.  170.  DOWNWARD- 
FLOW  TURBINES.  171.  IMPULSE  TURBINES.  172.  SPECIAL  DEVICES. 
173.  THE  NIAGARA  TURBINES 437-473 

CHAPTER  XV.     NAVAL  HYDROMECHANICS 

ART.  174.  GENERAL  PRINCIPLES.  175.  FRICTIONAL  RESISTANCE. 
176.  WORK  FOR  PROPULSION.  177.  THE  JET  PROPELLER.  178.  PADDLE 
WHEELS.  179.  THE  SCREW  PROPELLER.  ISO.  STABILITY  OF  A  SHIP. 
181.  ACTION  OF  THE  RUDDER.  182.  TIDES  AND  WAVES.  ,  474-494 


viii  CONTENTS 

CHAPTER  XVI.     PUMPS  AND  PUMPING 

PAGES 

ART.  183.  GENERAL  NOTES  AND  PRINCIPLES.  184.  RAISING  WATER 
BY  SUCTION.  185.  THE  FORCE  PUMP.  186.  LOSSES  IN  THE  FORCE 
PUMP.  187.  PUMPING  ENGINES.  188.  THE  CENTRIFUGAL  PUMP. 
189.  THE  HYDRAULIC  RAM.  190.  OTHER  KINDS  OP  PUMPS.  191.  PUMP- 
ING THROUGH  PIPES.  192.  PUMPING  THROUGH  HOSE 495-533 

APPENDIX 

ART.  193.  HYDRAULIC-ELECTRIC  ANALOGIES.  194.  PROBLEMS.  195. 
ANSWERS  TO  PROBLEMS.  196.  EXPLANATION  OF  TABLES.  .  .  .  534-542 

TABLES  FOR  ENGLISH  MEASURES 

TABLE  1.  HYDRAULIC  CONSTANTS.  3.  ENGLISH  UNITS  TO  METRIC 
EQUIVALENTS.  5.  INCHES  AND  FEET.  6.  GALLONS  AND  CUBIC  FEET. 
7.  WEIGHT  OF  DISTILLED  WATER.  9.  ATMOSPHERIC  PRESSURE.  11. 
ACCELERATION  CONSTANTS.  13.  HEADS  AND  PRESSURES.  15.  VELOC- 
ITIES AND  VELOCITY-HEADS.  17.  CIRCULAR  VERTICAL  ORIFICES. 
19.  SQUARE  VERTICAL  ORIFICES.  21.  RECTANGULAR  VERTICAL  ORI- 
FICES. 22.  SUBMERGED  ORIFICES.  23.  CONTRACTED  WEIRS.  25.  SUP- 
PRESSED WEIRS.  28.  WIDE-CRESTED  WEIRS.  29.  DAMS  32.  VER- 
TICAL JETS  FROM  NOZZLES.  33.  FRICTION  FACTORS  FOR  PIPES.  35. 
Loss  OF  HEAD  IN  PIPES.  37.  CIRCULAR  CONDUITS.  40.  RECTAN- 
GULAR CONDUITS.  42.  SEWERS.  44.  CHANNELS  IN  EARTH.  46.  BAZIN'S 
COEFFICIENTS  FOR  CHANNELS 543-565 

TABLES  FOR  METRIC  MEASURES 

TABLE  2.  HYDRAULIC  CONSTANTS.  4.  METRIC  UNITS  TO  ENGLISH 
EQUIVALENTS.  8.  WEIGHT  OF  DISTILLED  WATER.  10.  ATMOSPHERIC 
PRESSURE.  12.  ACCELERATION  CONSTANTS.  14.  HEADS  AND  PRES- 
SURES. 16.  VELOCITIES  AND  VELOCITY-HEADS.  18.  CIRCULAR  VER- 
TICAL ORIFICES.  20.  SQUARE  VERTICAL  ORIFICES.  24.  CONTRACTED 
WEIRS.  26.  SUPPRESSED  WEIRS.  30.  DAMS.  34.  FRICTION  FACTORS 
FOR  PIPES.  36.  Loss  OF  HEAD  IN  PIPES.  38.  CIRCULAR  CONDUITS. 
41.  RECTANGULAR  CONDUITS.  43.  SEWERS.  45.  CHANNELS  IN  E^ARTH. 
47.  BAZIN'S  COEFFICIENTS  FOR  CHANNELS 543-565 

TABLES  FOR  ALL  MEASURES 

TABLE  27.  SUBMERGED  WEIRS.  31.  CONICAL  TUBES.  39.  CIR- 
CULAR CONDUITS,  PARTLY  FULL.  48.  THE  BACKWATER  FUNCTION. 
49.  THE  DROP-DOWN  FUNCTION.  50.  SQUARES  OF  NUMBERS.  51. 
AREAS  OF  CIRCLES.  52.  TRIGONOMETRIC  FUNCTIONS.  53.  LOGARITHMS 
OF  TRIGONOMETRIC  FUNCTIONS.  54.  LOGARITHMS  OF  NUMBERS.  55. 
MATHEMATICAL  CONSTANTS 556-576 

INDEX   ,  577-585 


TREATISE  ON  HYDRAULICS 


CHAPTER  I 
FUNDAMENTAL  DATA 

ART.  1.     HISTORICAL  NOTES 

Hydraulics  is  that  branch  of  the  mechanics  of  fluids 
which  treats  of  water  in  motion,  while  Hydrostatics  treats 
of  water  at  rest.  These  two  branches  are  sometimes 
regarded  as  a  part  of  Hydromechanics,  the  name  of  the 
mechanics  of  fluids  and  gases.  While  the  main  purpose 
of  this  book  is  to  treat  of  water  in  motion,  the  most  im- 
portant principles  of  hydrostatics  will  also  be  discussed, 
since  these  are  necessary  for  a  complete  development  of 
the  laws  of  flow.  The  word  Hydraulics  is  hence  here  used 
as  closely  synonymous  with  the  hydromechanics  of  water. 

Hydraulics  is  a  modern  science  which  is  still  far  from 
perfect.  Archimedes,  about  250  B.C.,  established  a  few 
of  the  principles  of  hydrostatics  and  showed  that  the 
weight  of  an  immersed  body  was  less  than  its  weight  in 
air  by  the  weight  of  the  water  that  it  displaces.  Chain 
and  bucket  pumps  were  used  at  this  period  by  the  Egyp- 
tians, and  the  force  pump  was  invented  by  Ctesibius  about 
120  B.C.  The  Romans  built  aqueducts  as  early  as  300 
B.C.,  and  later  used  earthen  and  lead  pipes  to  convey  water 
from  them  to  their  houses.  They  knew  that  water  would 
rise  in  such  a  pipe  to  the  same  level  as  in  the  aqueduct 


2  FUNDAMENTAL  DATA  CHAP,  i 

and  that  a  slope  was  necessary  to  cause  flow  in  the  latter, 
but  had  no  conception  of  such  a  simple  quantity  as  a 
cubic  foot  per  minute.  But  after  the  destruction  of  Rome, 
in  475  A-D->  even  this  slight  knowledge  was  lost  and  Europe, 
for  a  thousand  years  sunk  in  barbarism,  made  no  scientific 
inquiries  until  the  Renaissance  period  began. 

Galileo,  in  1630,  studied  the  subject  of  the  flotation  of 
bodies  in  water,  and  a  little  later  his  pupils  Castelli  and 
Torricelli  made  notable  discoveries,  the  former  on  the 
flow  of  water  in  rivers  and  the  latter  on  the  height  of  a 
jet  issuing  from  an  orifice.  Pascal,  about  1650,  extended 
Torricelli 's  researches  on  the  influence  of  atmospheric 
pressure  in  causing  liquids  to  rise  in  a  vacuum.  Mario tte, 
about  1680,  was  the  first  to  consider  the  influence  of  fric- 
tion in  retarding  the  flow  in  pipes  and  channels,  and  Newton, 
in  1685,  was  the  first  to  note  the  contraction  of  a  jet  issuing 
from  an  orifice. 

During  the  eighteenth  century  notable  advances  were 
made.  >  Daniel  and  John  Bernoulli  extended  the  theory 
of  the  equilibrium  and  motion  of  fluids,  and  this  theory 
was  much  improved  and  generalized  by  D'AlemberL 
Bossut  and  Dubuat  made  experiments  on  the  flow  of  water 
in  pipes  and  deduced  practical  coefficients,  while  Chezy 
and  Prony,  near  the  close  of  the  century,  established 
general  formulas  for  computing  velocity  and  discharge. 

During  the  nineteenth  century  progress  in  every  branch 
of  hydraulics  was  great  and  rapid.  Eytelwein,  Weisbach, 
and  Hagen  stood  high  among  German  experimenters'; 
Venturi  and  Bidone  among  those  of  Italy ;  Poncelet,  Darcy, 
and  Bazin  among  those  of  France ;  while  Kutter  in  Switzer- 
land, Rankine  in  England,  and  James  B.  Francis  and 
Hamilton  Smith  in  America  also  took  high  rank  for  either 
practical  or  theoretical  investigations.  By  the  experiments 
and  discussions  of  these  and  many  other  engineers  the 
necessary  coefficients  for  the  discussion  of  orifices,  weirs> 


ART.  2  UNITS  OF  MEASURE  3 

jets,  pipes,  conduits,  "and  rivers  have  been  determined  and 
the  theory  of  the  flow  of  water  has  been  much  extended 
and  perfected.  The  invention  of  the  turbine  by  Fourneyron 
in  1827  exerted  much  influence  upon  the  development  of 
water  power,  while  the  studies  necessary  for  the  construc- 
tion of  canals  and  for  the  improvement  of  rivers  and  har- 
bors have  greatly  promoted  hydraulic  science.  In  this 
advance  the  engineers  of  the  United  States  have  done  good 
work  during  the  latter  part  of  the  nineteenth  century,  as 
is  shown  by  the  numerous  valuable  papers  published  in 
the  Transactions  of  the  American  engineering  societies, 
many  of  which  will  be  cited  in  the  following  pages. 

Galileo  said  in  1630  that  the  laws  controlling  the  motion 
of  the  planets  in  their  celestial  orbits  were  better  understood 
than  those  governing  the  motion  of  water  on  the  surface 
of  the  earth.  This  is  true  to-day,  for  the  theory  of  the  flow 
of  water  in  pipes  and  channels  has  not  yet  been  perfected. 
Experiment  is  now  in  advance  of  theory,  but  it  is  the  pur- 
pose of  the  author  to  present  both  in  this  volume  as  far 
as  practicable,  for  each  is  necessary  to  a  satisfactory  under- 
standing of  the  other. 

Prob.  1.  Rankine  published  in  the  Philosophical  Maga- 
zine of  September,  1858,  the  following  anagram  of  219  letters 
which  contains  his  discovery  regarding  the  hydraulic  resistance 
to  the  motion  of  a  boat:  2oa,  46,  6c,  gd,  34?,  8/,  4g,  i6h,  loi, 
5/f  3w,  isw,  140,  4P,  sq,  nr,  135,  25*,  4^,  2V,  2W,  ix,  $y.  What 
is  its  meaning? 

ART.  2.     UNITS  OF  MEASURE 

The  unit  of  linear  measure  universally  used  in  English 
and  American  hydraulic  literature  is  the  foot,  which  is  de- 
fined as  one -third  of  the  standard  yard.  For  some  minor 
purposes,  such  as  the  designation  of  the  diameters  of  orifices 
and  pipes,  the  inch  is  employed,  but  inches  should  always 
be  reduced  to  feet  for  use  in  hydraulic  formulas.  The  unit 
of  superficial  measure  is  usually  the  square  foot,  except 


4  FUNDAMENTAL  DATA  CHAP,  i 

for  the  expression  of  the  intensity  of  pressures,  when  the 
sqxiare  inch  is  more  commonly  employed. 

The  units  of  volume  employed  in  measuring  water  are 
the  cubic  foot  and  the  gallon,  but  the  latter  must  always 
be  reduced  to  cubic  feet  for  use  in  hydraulic  formulas. 
In  Great  Britain  and  its  colonies  the  Imperial  gallon  is 
used,  but  in  the  United  States  the  old  English  gallon  has 
continued  to  be  employed,  and  the  former  is  20  percent 
larger  than  the  latter.  The  following  are  the  relations 
between  the  cubic  foot  and  the  two  gallons : 

i  cubic  foot  =6.232  Imp.  gallons  =  7. 48 1  U.  S.  gallons; 
i  Imp.  gallon  =  o.i6o5  cubic  feet  =1.200  U.  S.  gallons; 
i  U.  S.  gallon  =  0.1 33 7  cubic  feet  =0.8331  Imp.  gallons; 

In  this  book  the  word  gallon  will  always  mean  the  United 
States  gallon  of  231  cubic  inches,  unless  otherwise  stated. 

The  unit  of  force  is  the  force  exerted  by  gravity  on  the 
avoirdupois  pound,  which  is  also  the  unit  for  measuring 
weights  and  pressures  of  water.  The  intensity  of  pressure 
is  measured  in  pounds  per  square  foot  or  in  pounds  per 
square  inch,  as  may  be  most  convenient,  and  sometimes  in 
atmospheres.  Gages  for  recording  the  pressure  of  water 
are  usually  graduated  to  read  pounds  per  square  inch. 

The  unit  of  time  to  be  used  in  all  hydraulic  formulas  is 
the  second,  although  in  numerical  problems  the  time  is 
often  stated  in  minutes,  hours,  or  days.  Velocity  is  de- 
nned as  the  space  passed  over  by  a  body  in  one  second,  under 
the  condition  of  uniform  motion,  so  that  velocities  are  to  be 
always  expressed  in  feet  per  second,  or  are  to  be  reduced 
to  these  units  if  stated  in  miles  per  hour  or  otherwise. 
Acceleration  is  the  velocity  gained  in  one  second,  and  it  is 
measured  in  feet  per  second  per  second. 

The  unit  of  work  is  the  foot-pound ;  that  is,  one  pound 
lifted  'through  a  vertical  distance  of  one  foot.  Energy  is 
work  which  can  be  done ;  for  example,  a  moving  stream  of 


ART.  3  PHYSICAL  PROPERTIES  OF  WATER  5 

water  has  the  ability  to  do  a  certain  amount  of  work  by 
virtue  of  the  quantity  of  moving  water  and  its  velocity,  and 
this  is  called  kinetic  energy.  Again,  a  quantity  of  water 
at  the  top  of  a  fall  has  the  ability  to  do  a  certain  amount 
of  work  by  virtue  of  its  height  above  the  foot  of  the  fall, 
and  this  is  called  potential  energy.  Potential  energy 
changes  into  kinetic  energy  as  the  water  drops,  and  kinetic 
energy  is  either  changed  into  heat  or  is  transformed,  by 
means  of  a  water  motor,  into  useful  work.  Power  is  work 
or  energy,  done  or  existing  in  a  specified  time,  and  the 
unit  for  its  measure  is  the  horse-power,  which  is  550  foot- 
pounds per  second. 

In  French  and  German  literature  the  metric  system  of 
measure  is  employed  and  this  is  far  more  convenient  than 
the  English  one  in  hydraulic  computations.  This  system 
is  understood  and  more  or  less  used  in  all  countries,  and 
its  universal  adoption  is  sure  to  occur  during  the  present 
century,  but  the  time  has  not  yet  come  when  an  American 
engineering  book  can  be  prepared  wholly  in  metric  meas- 
ures. This  treatise  will,  therefore,  mainly  use  the  English 
units  described  above,  but  at  the  close  of  most  of  the 
chapters  hydraulic  data  and  empirical  formulas  will  be 
given  in  metric  measures.  At  the  end  of  the  volume  the 
most  important  tables  will  be  found  in  both  systems. 
Tables  1  and  2  give  the  fundamental  hydraulic  constants, 
while  Tables  3  and  4  show  the  equivalents  in  each  system 
of  the  principal  units  in  the  other  system. 

Prob.  2.  If  1600  pounds  of  water  fall  every  second  from  a 
height  of  1 6  feet,  what  is  the  greatest  horse-power  that  can 
be  developed  by  the  stream? 

ART.  3.     PHYSICAL  PROPERTIES  OF  WATER 

At  ordinary  temperatures  pure  water  is  a  colorless  liquid 
which  possesses  perfect  fluidity;  that  is,  its  particles  have 
the  capacity  of  moving  over  each  other,  so  that  the  slightest 


6  FUNDAMENTAL  DATA  CHAP.  1 

disturbance  of  equilibrium   causes  a  flow.     It  is  a   conse- 
quence of  this  property  that  the  surface  of  still  water  is 

always    level;    also,   if 
several  vessels  or  tubes 
be  connected,  as  in  Fig. 
,  f^          |     3,  and  water  be  poured 
\-/  into   one    of    them,    it 

rises  in  the  others  until, 

when  equilibrium  ensues,  the  free  surfaces  are  in  the  same 
level  plane. 

The  free  surface  of  water  is  in  a  different  molecular 
condition  from  the  other  portions,  its  particles  being  drawn 
together  by  stronger  attractive  forces,  so  as  to  form  what 
may  be  called  the  "  skin  of  the  water,"  upon  which  insects 
walk.  The  skin  is  not  immediately  pierced  by  a  sharp 
point  which  moves  slowly  upward  toward  it,  but  a  slight 
elevation  occurs,  and  this  property  enables  precise  determi- 
nations of  the  level  of  still  water  to  be  made  by  means  of 
the  hook  gage  (Art.  35). 

At  about  32  degrees  Fahrenheit  a  great  alteration  in  the 
molecular  constitution  of  water  occurs,  and  ice  is  formed. 
If  a  quantity  of  water  be  kept  in  a  perfectly  quiet  condition, 
it  is  found  that  its  temperature  can  be  reduced  to  20°  or 
even  to  15°  Fahrenheit,  before  congelation  takes  place, 
but  at  the  moment  when  this  occurs  the  temperature  rises 
to  32°.  The  freezing-point  is  hence  not  constant,  but  the 
melting-point  of  ice  is  always  at  the  same  temperature  of 
32°  Fahrenheit  or  o°  centigrade. 

Ice  being  lighter  than  water,  forms  as  a  rule  upon  its 
surface ;  but  when  water  is  in  rapid  motion  a  variety  called 
anchor  ice  may  occur.  In  this  case  the  ice  is  formed  at  the 
surface  in  the  shape  of  small  needles,  which  are  quickly 
carried  to  the  lower  strata  by  the  agitation  due  to  the 
motion ;  there  the  needles  adhere  to  the  bed  of  the  stream, 
sometimes  accumulating  so  as  to  raise  the  water  level 


ART.  4  THE  WEIGHT  OF  WATER  7 

several  feet.*  Anchor  ice  sometimes  causes  obstruction  in 
conduits  leading  to  water  motors  or  even  clogs  and  stops 
the  motors  themselves. 

Water  is  a  solvent  of  high  efficiency,  and  is  therefore 
never  found  pure  in  nature.  Descending  in  the  form  of  rain 
it  absorbs  dust  and  gaseous  impurities  from  the  atmosphere ; 
flowing  over  the  surface  of  the  earth  it  absorbs  organic 
and  mineral  substances.  These  affect  its  weight  only 
slightly  as  long  as  it  remains  fresh,  but  when  it  has  reached 
the  sea  and  become  salt  its  weight  is  increased  more  than 
two  percent.  The  flow  of  water  through  orifices  is  only  in 
a  very  slight  degree  affected  by  the  impurities  held  in 
solution,  but  in  the  flow  through  pipes  they  often  cause 
incrustation  or  corrosion  which  increases  the  roughness  of 
the  surface  and  diminishes  the  velocity. 

The  capacity  of  water  for  heat,  the  latent  heat  evolved 
when  it  freezes,  and  that  absorbed  when  it  is  transformed 
into  steam,  need  not  be  considered  for  the  purposes  of 
hydraulic  investigations.  Other  physical  properties,  such 
as  its  variation  in  volume  with  the  temperature,  its  com- 
pressibility, and  its  capacity  for  transmitting  pressures,  are 
discussed  in  the  following  pages.  The  laws  which  govern 
its  pressure,  flow,  and  energy  under  various  circumstances 
belong  to  the  science  of  Hydraulics,  and  form  the  subject- 
matter  of  this  volume. 

Prob.  3.  How  many  degrees  centigrade  are  equivalent  to 
15°  Fahrenheit?  How  many  degrees  Fahrenheit  are  equiv- 
alent to  25°  centigrade? 

ART.  4.     THE  WEIGHT  OF  WATER 

The  weight  of  water  per  unit  of  volume  depends  upon 
the  temperature  and  upon  its  degree  of  purity.  The 

*  Francis,  Transactions  of  the  American  Society  of  Civil  Engineers,  1884, 
vol.  10,  p.  192. 


8  FUNDAMENTAL  DATA  CHAP.  I 

following  approximate  values  are,  however,  those  generally 
employed  except  when  great  precision  is  required: 

i  cubic  foot  weighs  62.5      pounds, 
i  U.S.  gallon  weighs  8.355  pounds. 

These  values  will  be  used  in  this  book,  unless  otherwise 
stated,  in  the  solution  of  the  examples  and  problems. 

The  weight  per  unit  of  volume  of  pure  distilled  water  is 
the  greatest  at  the  temperature  of  its  maximum  density, 
3 9°. 3  Fahrenheit,  and  least  at  the  boiling-point.  For 
ordinary  computations  the  variation  in  weight  due  to 
temperature  is  not  considered,  but  in  tests  of  the  efficiency 
of  hydraulic  motors  and  of  pumps  it  should  be  regarded. 
Table  7  at  the  end  of  this  book  contains  the  weights  of  one 
cubic  foot  of-  pure  water  at  different  temperatures  as  de- 
duced by  Smith  from  the  experiments  of  Rossetti.* 

Waters  of  rivers,  springs,  and  lakes  hold  m  suspension 
and  solution  inorganic  matters  which  cause  the  weight  per 
unit  of  volume  to  be  slightly  greater  than  for  pure  water. 
River  waters  are  usually  between  62.3  and  62.6  pounds  per 
cubic  foot,  depending  upon  the  amount  of  impurities  and 
on  the  temperature,  while  the  water  of  some  mineral  springs 
has  been  found  to  be  as  high  as  62.7.  It  appears  that,  in 
the  absence  of  specific  information  regarding  a  particular 
water,  the  weight  62.5  pounds  per  cubic  foot  is  a  fair 
approximate  value  to  use.  It  also  has  the  advantage 
of  being  a  convenient  number  in  computations,  for  62.5 
pounds  is  1000  ounces,  or  -f$-  is  the  equivalent  of  62.5. 

Brackish  and  salt  waters  are  always  much  heavier  than 
fresh. water.  For  the  Gulf  of  Mexico  the  weight  per  cubic 
foot  is  about  63.9,  for  the  oceans  about  64.1,  while  for  the 
Dead  Sea  there  is  stated  the  value  73  pounds  per  cubic  foot. 

*  Hamilton  Smith,  Jr.,  Hydraulics:  The  Flow  of  Water  through 
Orifices  over.  Weirs,  and  through  open  Conduits  and  Pipes  (London  and 
New  York,  1886),  p.  14. 


ART.  5  ATMOSPHERIC  PRESSURE  9 

The  weight  of  ice  per  cubic  foot  varies  from  57.2  to  57.5 
pounds.  The  sewage  of  American  cities  is  impure  water 
which  weighs  from  62.4  to  62.7  pounds  per  cubic  foot,  but 
the  sewage  of  European  cities  is  somewhat  heavier  on 
account  of  the  smaller  amount  of  water  that  is  turned 
into  the  sewers. 

Prob.  4.  How  many  gallons  of  water  are  contained  in  a 
pipe  6  inches  in  diameter  and  12  feet  long?  How  many  pounds? 

ART.  5.     ATMOSPHERIC  PRESSURE 

Torricelli  in  1643  discovered  that  the  atmospheric 
pressure  would  cause  mercury  to  rise  in  a  tube  from  which 
the  air  had  been  exhausted.  This  instrument  is  called  the 
mercury  barometer,  and  owing  to  the  great  density  of 
mercury  the  height  of  the  column  required  to  balance  the 
atmospheric  pressure  is  only  about  30  inches.  When 
water  is  used  in  the  vacuum  tube  the  height  of  the  column 
is  about  34  feet.  In  both  cases  the  weight  of  the  barometric 
column  is  equal  to  the  weight  of  a  column  of  air  of  the 
same  cross-section  as  that  of  the  tube,  both  columns  being 
measured  upward  from  the  common  surface  of  contact. 

The  atmosphere  exerts  its  pressure  with  varying  inten- 
sity as  indicated  by  the  readings  of  the  mercury  barometer. 
At  and  near  the  sea  level  the  average  reading  is  30  inches, 
and  as  mercury  weighs  0.49  pounds  per  cubic  inch  at  com- 
mon temperatures,  the  average  atmospheric  pressure  is 
taken  to  be  30X0.49  or  14.7  pounds  per  square  inch.  The 
pressure  of  one  atmosphere  is  therefore  defined  to  be  a 
pressure  of  14.7  pounds  per  square  inch.  Then  a  pressure 
of  two  atmospheres  is  29.4  pounds  per  square  inch.  And 
conversely,  a  pressure  of  100  pounds  per  square  inch  may 
be  expressed  as  a  pressure  of  6.8  atmospheres. 

Pascal  in  1646  carried  a  mercury  barometer  to  the  top 
of  a  mountain  and  found  that  the  height  of  the  mercury 


10  FUNDAMENTAL  DATA  CHAP.  I 

column  decreased  as  he  ascended.  It  was  thus  definitely 
proved  that  the  cause  of  the  ascent  of  the  liquid  in  the 
vacuum  tube  was  due  to  the  pressure  of  the  air.  Since 
mercury  is  13.6  times  heavier  than  water  a  column  of  water 
should  rise  to  a  height  of  30X13.6=408  inches  =  34  feet 
under  the  pressure  of  one  atmosphere,  and  this  was  also 
found  to  be  the  case.  A  water  barometer  is  impracticable 
for  use  in  measuring  atmospheric  pressures,  but  it  is  con- 
venient to  know  its  approximate  height  corresponding  to 
a  given  height  of  the  mercury  barometer.  Table  9  at  the 
end  of  this  volume  shows  heights  of  the  mercury  and  water 
barometers,  with  the  corresponding  pressures  in  pounds  per 
square  inch  and  in  atmospheres.  It  also  gives,  in  the  fifth 
column,  values  from  the  vertical  scale  of  altitudes  used  in 
barometric  leveling  which  show  approximate  elevations 
above  sea  level  corresponding  to  barometer  readings, 
provided  that  the  reading  at  sea  level  is  30  inches.*  In  the 
last  column  are  approximate  boiling-points  of  water  cor- 
responding to  the  readings  of  the  mercury  barometer. 

The  atmospheric  pressure  must  be  taken  into  account 
in  many  computations  on  the  flow  of  water  in  tubes  and 
pipes.  It  is  this  pressure  that  causes  water  to  flow  in 
syphons  and  to  rise  in  tubes  from  which  the  air  has  been 
exhausted.  By  virtue  of  this  pressure  the  suction  pump 
is  rendered  possible,  and  all  forms  of  injector  pumps  depend 
upon  it  to  a  certain  degree.  On  a  planet  without  an  atmos- 
phere many  of  the  phenomena  of  hydraulics  would  be  quite 
different  from  those  observed  on  this  earth. 

Prob.  5a.  Find  from  Table  9  the  height  of  the  mercury 
barometer  when  the  boiling-point  of  water  is  205°  Fahrenheit. 

Prob.  5b.  A  mercury  barometer  reads  29.66  inches  at  the 
foot  of  a  hill,  and  at  the  same  time  another  barometer  reads 
28.56  inches  at  the  top.  What  is  the  difference  in  height  be- 
tween the  two  stations? 

*  Plympton,  The  Aneroid  Barometer  (New  York,  1885),  p.  88. 


ART.  6  COMPRESSIBILITY  OF  WATER  11 


ART.  6.     COMPRESSIBILITY  OF  WATER 

The  popular  opinion  that  water  is  incompressible  .s  not 
justified  by  experiments,  which  show  in  fact  that  it  is  more 
compressible  than  iron  or  even  timber  within  the  elastic 
limit.  These  experiments  indicate  that  the  amount  of 
compression  is  directly  proportional  to  the  applied  pressure, 
and  that  water  is  perfectly  elastic,  recovering  its  original 
form  on  the  removal  of  the  pressure.  The  decrease  in  the 
unit  of  volume  caused  by  a  pressure  of  one  atmosphere 
varies,  according  to  the  experiments  of  Grassi,  from  0.000051 
at  35°  Fahrenheit  to  0.000045  at  8°°  Fahrenheit.*  As  a 
mean  value  0.00005  may  be  taken  for  this  cubical  unit- 
compression. 

A  vertical  column  of  water  accordingly  increases  in 
density  from  the  surface  downward.  If  its  weight  at  the 
surface  be  62.5  pounds  per  cubic  foot,  at  a  depth  of  34 
feet  the  weight  of  a  cubic  foot  will  be 

62.5(1  +0.00005)  =62.503  pounds, 
and  at  a  depth  of  340  feet  a  cubic  foot  will  weigh 
62.5(1  +0.0005)  =  62.53  pounds. 

The  variation  in  weight,  due  to  compressibility,  is  hence  too 
small  to  be  regarded  in  hydrostatic  computations. 

The  modulus  of  elasticity  of  volume  for  water  is  the 
ratio  of  the  unit-stress  to  the  cubical  unit -compression,  or 

E= — '-  -^—  =294  ooo  pounds  per  square  inch. 
0.00005 

The  modulus  of  elasticity  of  volume  for  steel,  when  subjected 
to  uniform  hydrostatic  pressure,  is  the  same  as  the  com- 
mon modulus  due  to  stress  in  one  direction  only,  or  E  = 
30  ooo  ooo  pounds  per  square  inch.  Hence  water  is  about 
100  times  more  compressible  than  steel. 

*  Grassi,  Annales  de  chemie  et  physique,  1851,  vol.  31,  p.  437. 


12  FUNDAMENTAL  DATA  CHAP.  I 

The  velocity  of  sound  or  stress  in  any  substance  is  given 
by  the  formula  u  =  \/Eg/w,  where  w  is  the  weight  of  a 
cubic  unit  of  the  material  weighed  by  a  spring  balance  at 
the  place  where  the  acceleration  of  gravity  is  g  (Art.  7). 
For  water  having  1^  =  62.4  pounds  per  cubic  foot  at  a  place 
where  g  =  32 . 2  feet  per  second  per  second,  and  E  =  42  300  ooo 
pounds  per  square  foot,  this  formula  gives  24  =  4670  feet 
per  second  for  the  velocity  of  sound,  which  agrees  well 
with  the  results  of  experiments. 

In  order  to  deduce  the  above  formula  for  the  velocity 
of  stress  it  is  necessary  to  use  some  of  the  fundamental 
principles  of  elementary  mechanics  and  of  the  mechanics 
of  elastic  bodies.  Let  a  free  rigid  body  of  weight  W  be 
acted  upon  for  one  second  by  a  constant  force  F  and  let 
/  be  the  velocity  of  the  body  at  the  end  of  one  second. 
Let  g  be  the  velocity  gained  in  one  second  by  W  when  falling 
under  the  action  of  the  constant  force  of  gravity.  Then, 
since  forces  are  proportional  to  their  accelerations, 

F/W-f/g        or        F-W.f/g 

and  during  the  second  of  time  the  body  has  moved  the 
distance  J/.  Now,  consider  a  long  elastic  bar  of  the  length 
u,  so  that  a  force  applied  at  one  end  will  be  felt  at  the  other 
end  in  one  second,  it  being  propagated  by  virtue  of  the 
elasticity  of  the  material.  Let  A  be  the  area  of  the  cross- 
section  of  the  bar  and  E  the  modulus  or  coefficient  of 
elasticity  of  the  material.  When  a  constant  compressive 
force  jp  is  applied  to  the  bar  the  shortening  ultimately 
produced  is  zFu/AE*  but  if  this  be  done  for  one  second 
only  the  elongation  is  only  half  this  amount,  since  the  first 
increment  of  stress  is  just  reaching  the  other  end  of  the  bar 
at  the  end  of  the  second.  The  center  of  gravity  of  the  bar 
has  then  moved  through  the  distance  ^Fu/AE,  and  its 
velocity  v  is  Fu/AE.  If  w  be  the  weight  of  a  cubic  unit  of 

*Merriman's  Mechanics  of  Materials  (New  York,  1902),  pp.  13,  197. 


ART.  7  ACCELERATION  DUE  TO  GRAVITY  13 

the  material,  the  weight  W  is  wAu.     Inserting  these  values 
of  v  and  W  in  the  above  equation,  there  is  found 

F         Fu  \Eg 

— ^r— =   ,  F  whence         ^=Nj— -  (6) 

wAu      AEg  >*  w 

which  is  the  formula  for  the  propagation  of  sound  or  stress 
in  elastic  materials  first  established  by  Newton. 

Prob.  6a.  Compute  the  velocity  of  sound  in  distilled  water 
at  35°  and  also  at  80°  Fahrenheit. 

Prob.  66.  If  the  weight  of  a  cubic  foot  of  water  at  the  sur- 
face of  the  ocean  is  64.3  pounds,  what  is  the  weight  of  a  cubic 
foot  at  the  depth  of  two  miles? 

ART.  7.     ACCELERATION  DUE  TO  GRAVITY 

The  motion  of  water  in  river  channels,,  and  its  flow 
through  orifices  and  pipes,  is  produced  by  the  force  of  gravity. 
This  force  is  proportional  to  the  acceleration  of  the  velocity 
of  a  body  falling  freely  in  a  vacuum ;  that  is,  to  the  increase 
in  velocity  in  one  seetfnd.  Acceleration  is  measured  in  feet 
per  second  per  second,  so  that  its  numerical  value  represents 
the  number  of  feet  per  second  which  have  been  gained  in 
one  second.  The  letter  g  is  used  to  denote  the  acceleration 
of  a  falling  body  near  the  surface  of  the  earth.  In  pure 
mechanics  g  is  found  in  all  formulas  relating  to  falling 
bodies;  for  instance,  if  a  body  falls  from  rest  through  the 
height  h  it  attains  in  a  vacuum  a  velocity  equal  to  V2gh. 
In  hydraulics  g  is  found  in  all  formulas  which  express  the 
laws  of  flow  of  water  under  the  influence  of  gravity. 

The  quantity  32.2  feet  per  second  per  second  is  an 
approximate  value  of  g  which  is  often  used  in  hydraulic 
formulas.  It  is,  however,  well  known  that  the  force  of 
gravity  is  not  of  constant  intensity  over  the  earth's  surface, 
but  is  greater  at  the  poles  than  at  the  equator,  and  also 
greater  at  the  sea  level  than  on  high  mountains.  The 


14  FUNDAMENTAL  DATA  CHAP.  I 

following  formula  of  Peirce,  which  is  partly  theoretical  and 
partly  empirical,  gives  g  in  feet  for  any  latitude  /,  and 
any  elevation  e  above  the  sea  level,  e  being  in  feet: 

g  =  32.  0894  (1+0.0052375  sin2/)  (1—0,00000009570)  (7)4 
and  from  this  its  value  may  be  computed  for  any  locality. 

The  greatest  value  of  g  is  at  the  sea  level  at  the  pole, 
and  for  this  locality  I  =  90°,  e  =  o,  whence  £  =  32.258.  The 
least  value  of  g  is  on  high  mountains  at  the  equator;  for 
this  there  may  be  taken  /=o°,  0  =  10000  feet,  whence 
£  =  32.059.  The  mean  of  these  is  the  value  of  the  accel- 
eration used  in  this  book,  unless  otherwise  stated,  namely, 

£  =  32.16  feet  per  second  per  second, 

and  from  this  the  mean  values  of  the  frequently  occurring 
quantities  \/2g  and  i/2g  are  found  to  be 


=  8.020,  I/2£=O.OI555.  (7), 

If  greater  precision  be  required,  which  will  sometimes  be  the 
case,  g  can  be  computed  from  the  above  formula  for  the 
particular  latitude  and  elevation.  Table  11  gives  multiples 
of  the  quantities  g,  2gt  i/2g  and  V/2£  which  will  often  be 
useful  in  numerical  computations. 

Prob.  7  a.  The  acceleration  of  gravity  on  the  planet  Mars 
is  12.2  feet  per  second  per  second  and  the  weight  of  a  cubic 
foot  of  water  is  23.7  pounds.  Compute  the  velocity  of  sound 
in  the  water  of  the  Martian  canals. 

Prob.  7b.  Compute  to  four  significant  figures  the  values  of 
g  and  \/2g  for  the  latitude  40°  36'  and  the  elevation  400  feet. 
Also  for  the  same  latitude  and  the  elevation  4000  feet. 

ART.  8.     NUMERICAL  COMPUTATIONS. 

The  numerical  work  of  computation  should  not  be 
carried  to  a  greater  degree  of  refinement  than  the  data  of 
the  problem  warrant.  For  instance,  in  questions  relating 


ART.  8  NUMERICAL  COMPUTATIONS  15 

to  pressures,  trie  data  are  uncertain  in  the  third  significant 
figure,  and  hence  more  figures  than  three  in  the  final  result 
must  be  delusive.  Thus,  let  it  be  required  to  compute  the 
number  of  pounds  of  water  in  a  box  containing  307.37 
cubic  feet.  Taking  the  mean  value  62.5  pounds  as  the 
weight  of  one  cubic  foot,  the  multiplication  gives  the  result 
19  210.625  pounds,  but  evidently  the  decimals  here  have  no 
precision,  since  the  last  figure  in  62.5  is  not  accurate,  and 
is  likely  to  be  less  than  5,  depending  upon  the  impurity 
of  the  water  and  its  temperature.  The  proper  answer  to 
this  problem  is  19  200  pounds,  or  perhaps  19  210  pounds, 
and  this  is  to  be  regarded  as  a  probable  average  result 
rather  than  an  exact  definite  quantity. 

Three  significant  figures  are  usually  sufficient  in  the 
answer  to  any  hydraulic. problem,  but  in  order  that  the 
last  one  may  be  correct  four  significant  figures  should  be 
used  in  the  computations.  Thus,  307.37  has  five  significant 
figures  and  this  should  be  written  307.4  before  multiplying 
it  by  62 . 5 .  The  zeros  following  a  decimal  point  of  a  decimal 
are  not  counted  significant  figures;  thus,  0.0019  nas  two 
and  0.0003742  has.  four  significant  figures. 

The  use  of  logarithms  is  to  be  recommended  in  hydraulic 
computations,  as  thereby  both  mental  labor  and  time  are 
saved.  Four-figure  tables  are  sufficient  for  common  prob- 
lems, and  their  use  is  particularly  advantageous  in  all 
cases  where  the  data  are  not  precise,  as  thus  the  number 
of  significant  figures  in  final  results  is  kept  at  about  three, 
and  hence  statements  implying  great  precision,  when  none 
really  exists,  are  prevented.  The  four-place  logarithmic 
table  at  the  end  of  this  volume  will  be  found  very  convenient 
in  solving  numerical  problems.  As  an  example,  let  it  be 
required  to  find  the  weight  of  a  column  of  water  2.66  inches 
square  and  28.7  feet  long.  The  computation,  both  by 
common  arithmetic  and  by  logarithms  is  as  follows,  and  it 
will  be  found,  by  trying  similar  problems,  that  in  general  the 


16  FUNDAMENTAL  DATA  CHAP,  i 

By  Arithmetic.  By  Logarithms, 


2.66          0.04914 
2.66  28.7 


5-32  9^28 

i  596  39312 

160)144          3439 
7.076(0.04914 
576 


1316 
1296 

14      Ans.  88.1  pounds. 


2.66   0.4249 

2 

0.8498 

144  2.1584 

2.6914 

28.7  1-4579 

62.5  1-7959 

Ans.  88.1  1.9452 


use  of  logarithms  effects  a  saving  of  time  and  labor.  The 
common  slide  rule,  which  is  constructed  on  the  logarithmic 
idea,  will  also  be  found  very  useful  in  the  numerical  work 
of  hydraulic  problems. 

The  tables  of  constants,  squares,  and  areas  of  circles 
at  the  end  of  this  volume  will  also  be  advantageous  in 
abridging  computations.  For  instance,  it  is  seen  at  once 
from  Table  50  that  the  square  of  2.66  to  four  significant 
figures  is  7.076,  while  Table  51  shows  that  the  area  of  a 
circle  having  a  diameter  of  0.543  inches  is  0.2316  square 
inches.  The  logarithms  of  hydraulic  and  mathematical 
constants  are  given  in  Tables  1,  2,  and  55.  Tables  5,  6,  11, 
12,  13,  14,  15,  and  16  give  multiples  of  constants  which  may 
be  advantageously  used  when  it  is  necessary  to  multiply 
several  numbers  by  the  same  constant.  For  example,  if 
it  be  required  to  reduce  333.4,  318.7,  and  98.6  cubic  feet 
to  U.  S.  gallons,  the  book  is  opened  at  Table  6  where  the 
multiples  of  7.481  are  given,  and  the  work  is  as  follows: 

318.7 
2244.2 

74.8 
59-8 


737-5 


2384.0 

These  results  are  more  accurate  than  can  be  obtained  with 
four-place  logarithmic  tables.  The  logarithmic  work  for 
this  case  would  be  the  following: 


ART.  8  NUMERICAL  COMPUTATIONS  17 

333-4  3i8.7  98.6 

2.5229  2.5034  1.9939 

0.8740  0.8740  0.8740 

3.3969  3-3774  2.8679 

2494  2384  737-7 

In  all  cases  it  is  desirable  that  computations  should  be 
checked,  and  a  good  method  of  doing  this  is  to  make  one 
computation  by  common  arithmetic  and  another  by 
logarithms. 

As  this  book  is  mainly  intended  for  the  use  of  students  in 
technical  schools,  a  word  of  advice  directed  especially  to 
them  may  not  be  inappropriate.  It  will  be  necessary  for 
students,  in  order  to  gain  a  clear  understanding  of  hydraulic 
science,  or  of  any  other  engineering  subject,  to  solve  many 
numerical  problems,  and  in  this  a  neat  and  systematic 
method  should  be  cultivated.  The  practice  of  performing 
computations  on  any  loose  scraps  of  paper  that  may  happen 
to  be  at  hand  should  be  at  once  discontinued  by  every 
student  who  has  followed  it,  and  he  should  hereafter  solve 
his  problems  in  a  special  book  provided  for  that  purpose, 
and  accompany  them  by  such  explanatory  remarks  as  may 
seem  necessary  in  order  to  render  the  solutions  clear. 
Such  a  note-book,  written  in  ink,  and  containing  the  fully 
worked  out  solutions  of  the  examples  and  problems  given 
in  these  pages,  will  prove  of  great  value  to  every  student 
who  makes  it.  Before  beginning  the  solution  of  a  problem, 
a  diagram  should  be  drawn  whenever  it  is  possible,  for  a 
diagram  helps  the  student  to  clearly  understand  the  problem, 
and  a  problem  thoroughly  understood  is  half  solved.  Be- 
fore commencing  the  numerical  work,  it  is  also  well  to  make 
a  mental  estimate  of  the  final  result. 

Prob.  8a.  Compute  the  diameter,  in  inches,  of  a  cylindrical 
column  of  water  34  feet  high  which  weighs  14.73  pounds  at 
the  temperature  of  62°  Fahrenheit. 

Prob.  86.  When  the  height  of  the  water  barometer  is  33.3 
feet,  what  is  the  height  of  the  mercury  barometer,  and  what 
is  the  atmospheric  pressure  in  pounds  per  square  inch? 


18  FUNDAMENTAL  DATA  CHAP,  i 

ART.  9.     DATA  IN  THE  METRIC  SYSTEM 

When  the  metric  system  is  used  for  hydraulic  computa- 
tions the  meter  is  taken  as  the  unit  of  length,  the  cubic 
meter  as  the  unit  of  volume,  and  the  kilogram  as  the  unit 
of  force  and  weight.  Lengths  are  sometimes  expressed  in 
centimeters  and  volumes  in  liters,  but  these  should  be 
reduced  to  meters  and  cubic  meters  for  use  in  the  formulas. 
The  unit  of  time  is  the  second,  the  unit  of  velocity  is  one 
meter  per  second,  and  accelerations  are  measured  in  meters 
per  second  per  second.  Pressures  are  usually  expressed 
in  kilograms  per  square  centimeter  and  densities  in  kilo- 
grams per  cubic  meter.  The  metric  horse-power  is  75 
kilogram -meters  of  work  per  second,  and  this  is  about  ij 
percent  less  than  the  English  horse-power.  Tables  3  and  4 
give  the  equivalents  in  each  system  of  the  units  of  the  other 
system,  but  the  student  will  rarely  need  to  use  these  tables. 
He  should,  on  the  other  hand,  when  using  the  metric 
system  employ  it  exclusively  and  learn  to  think  readily  in 
it.  The  following  matter  is  supplementary  to  the  cor- 
responding articles  of  the  preceding  pages. 

(Art.  3)  At  about  o°  centigrade  ice  is  generally  formed. 
If  water  be  kept  perfectly  quiet,  however,  it  is  found  that 
its  temperature  can  be  reduced  to  -7°  or  —9°  before 
freezing  begins,  but  at  this  instant  the  temperature  rises 
to  o°  centigrade. 

(Art.  4)  In  the  metric  system  the  following  approxi- 
mate values  are  used  for  the  weight  of  water : 

i   cubic  meter  weighs  1000  kilograms, 
i  liter  weighs  i  kilogram. 

It  may  be  noted  that  the  constants  for  the  weight  of  water 
differ  slightly  in  the  two  systems.  Thus,  the  equivalent  of 
62.5  pounds  per  cubic  foot  is  about  1001  kilograms  per 
cubic  meter.  The  weight  per  unit  of  volume  of  pure 
distilled  water  is  greatest  at  the  temperature  of  maximum 


ART.  9  DATA  IN  THE  METRIC  SYSTEM  19 

density,  4°.i  centigrade,  and  least  at  the  boiling-point. 
Table  8  gives  weights  of  distilled  water  at  different  tempera- 
tures in  kilograms  per  cubic  meter,  as  determined  by  Ros- 
setti.*  River  waters  are  usually  between  997  and  1001 
kilograms  per  cubic  meter,  depending  upon  the  amount  of 
impurities  and  the  temperature,  while  the  water  of  some 
mineral  springs  has  been  found  as  high  as  1004.  It  appears 
then  that  1000  kilograms  per  cubic  meter  is  a  fair  average 
value  to  use  in  hydraulic  work  for  the  weight  of  fresh  water. 
Brackish  and  salt  waters  are  heavier.  For  the  Gulf  of 
Mexico  the  weight  per  cubic  meter  is  about  1023,  for  the 
oceans  about  1027,  while  for  the  Dead  Sea  there  is  stated 
the  value  1169  kilograms  per  cubic  meter.  The  weight  of 
ice  per  cubic  meter  varies  from  916  to  921  kilograms. 

(Art.  5)  Near  the  sea  level  the  average  reading  of  the 
mercury  barometer  is  76  centimeters,  and,  since  mercury 
weighs  13.6  grams  per  cubic  centimeter,  the  average  at- 
mospheric pressure  is  taken  to  be  76X0.0136  =  1.0333 
kilograms  per  square  centimeter.  One  atmosphere  of 
pressure  is  therefore  slightly  greater  than  a  pressure  of 
one  kilogram  per  square  centimeter.  Conversely,  a  pressure 
of  one  kilogram  per  square  centimeter  may  be  expressed 
as  a  pressure  of  0.968  atmospheres.  In  a  perfect  vacuum 
water  will  rise  to  a  height  of  about  10^  meters  under  a 
mean  pressure  of  one  atmosphere,  for  the  average  specific 
gravity  of  mercury  is  13.6,  and  13.6X0.76  =  10.33  meters. 
Table  10  shows  atmospheric  pressures,  altitudes,  and  boil- 
ing-points of  water  corresponding  to  heights  of  the  mercury 
and  water  barometers. 

(Art.  6)  If  the  weight  of  a  cubic  meter  of  water  is 
1000  kilograms  at  the  surface  of  a  pond  the  weight  of  a 
cubic  meter  at  a  depth  of  loj  meters  will  be 

1000 (i  +0.00005)  =  1000.05  kilograms, 

*  Annales  de  chimie  et  de  physique,  1869,  vol.  17,  page  370. 


20  FUNDAMENTAL  DATA  CHAP.  I 

and  at  a  depth  of  103^  meters  a  cubic  meter  will  weigh 
i ooo ( i  +0.0005)  =  1000.5  kilograms. 

Hence  the  variation  due  to  compression  is  too. small  to  be 
generally  taken  into  account.  The  modulus  of  elasticity 
of  volume  for  water  is 

E  = — '          =  20  700  kilograms  per  square  centimeter, 
0.00005 

while  that  of  steel  is  about  2  100  ooo.  Using  g  =  9.8  meters 
per  second  per  second,  the  mean  velocity  of  sound  in 
water  is  v  =  \/Eg/w  =  14.20  meters  per  second. 

(Art.  7)  The  formula  of  Peirce  for  the  acceleration  of 
gravity  on  the  earth's  surface  is 

£  =  9.78085(1+0.0052375  sin2/) (1-0.0000003 140)     (9)t 

in  which  g  is  the  acceleration  in  meters  per  second  per 
second  at  a  place  whose  latitude  is  /  degrees  and  whose 
elevation  is  e  meters  above  the  sea  level.  The  greatest 
value  of  g  is  at  the  sea  level  at  the  pole ;  here  /  =  90°  and 
e  =  o,  whence  £  =  9.8322.  The  least  value  of  g  in  hydraulic 
practice  is  found  on  high  lands  at  the  equator;  here  Z  =  o° 
and  £  =  4000  meters,  whence  £  =  9.7683.  The  mean  of 
these  is  9.800,  which  closely  agrees  with  that  found  in  Art. 
7,  since  32.16  feet  equals  9.802  meters;  accordingly 

£  =  9.800  meters  per  second  per  second 

is  the  value  of  the  acceleration  that  will  be  used  in  the 
metric  work  of  this  book.  From  this  are  found 

\/2£  =  4.427  i/2£  =  0.05102          ;     (9)2 

Table  12  gives  multiples  of  these  values  which  will  often 
be  of  use  in  numerical  computations. 

(Art.  8)  The  remarks  as  to  precision  of  numerical 
computation  also  apply  here.  Thus,  if  it  be  required  to 
find  the  weight  of  water  in  a  pipe  38  centimeters  in  diameter 


ART.  9  DATA  IN  THE  METRIC  SYSTEM  21 

and  6  meters  long,  Table  51  gives  0.1134  square  meters  for 
the  sectional  area,  the  volume  is  then  0.6804  cubic  meters, 
and  the  weight  is  680  kilograms,  the  fourth  figure  being 
omitted  because  nothing  is  known  about  the  temperature 
or  purity  of  the  water.  In  general,  hydraulic  computations 
are  much  easier  in  the  metric  than  in  the  English  system. 

Prob.  9a.  What  is  the  pressure  in  kilograms  per  square 
centimeter  at  the  base  of  a  column  of  water  97.3  meters  high? 

Prob.  96.  Compute  the  value  of  g  at  Quito,  Ecuador,  which 
is  in  latitude  —  o°  13'  and  at  an  elevation  of  2850  meters 
above  sea  level. 

Prob.  9c.  Compute  the  velocity  of  sound  in  fresh  distilled 
water  at  the  temperature  of  12°  centigrade,  and  also  its  mean 
velocity  in  salt  water. 

Prob.  9d.  How  many  cubic  meters  of  water  are  contained 
in  a  pipe  3150  meters  long  and  30  centimeters  in  diameter? 
How  many  kilograms?  How  many  metric  tons? 

Prob.  9e.  What  is  the  boiling-point  of  water  when  the 
mercury  barometer  reads  735  millimeters?  How  high  will 
water  rise  in  a  vacuum  tube  at  a  place  where  the  boiling-point 
of  water  is  92°  centigrade? 


22 


HYDROSTATICS. 


CHAP.  II 


CHAPTER  II 


HYDROSTATICS 


ART.  10.     TRANSMISSION  OF  PRESSURE 

One  of  the  most  remarkable  properties  of  a  fluid  is  its 
capacity  of  transmitting  a  pressure,  applied  at  one  point  of 
the  surface  of  a  closed  vessel,  unchanged  in  intensity,  in  all 
directions,  so  that  the  effect  of  the  applied  pressure  is  to 
cause  an  equal  force  per  square  inch  upon  all  parts  of  the 
enclosing  surface.  Pascal,  in  1646,  was  the  first  to  note 
that  great  forces  could  be  produced  in  this  manner;  he 

saw  that  the  total  pressure  in- 
creased proportionally  with  the 
area  of  the  surface.  Taking  a 
closed  barrel  filled  with  water 
he  inserted  a  small  vertical  tube 
of  considerable  length  tightly 
into  it  and  on  filling  the  tube 
the  barrel  burst  under  the  great 
pressure  thus  produced  on  its  sides,  although  the  weight  of 
the  water  in  the  tube  was  quite  small.  The  first  diagram 
in  Fig.  10a  represents  Pascal's  barrel,  and  it  is  seen  that 
the  unit  -pressure  at  B  is  due  to  the  head  AB  and  inde- 
pendent of  the  size  of  the  tube  AC. 

Pascal  clearly  saw  that  this  property  of  water  could  be 
employed  in  a  useful  manner  in  mechanics,  but  it  was  not 
until  1796  that  Bramah  built  the  first  successful  hydraulic 
press.  This  machine  has  two  pistons  of  different  sizes  and 
a  force  applied  to  the  small  piston  is  transmitted  through 
the  fluid  and  produces  an  equal  unit-pressure  at  every 


-  10a 


ART.  10 


TRANSMISSION  OF  PRESSURE 


23 


FIG.  106 


point  on  the  large  piston.  The  applied  force  is  here  multi- 
plied to  any  required  extent,  but  the  work  performed  by 
the  large  piston  cannot  ex- 
ceed that  imparted  to  the 
•fluid  by  the  small  one.  Let 
a  and  A  be  the  areas  of  the 
small  and  large  pistons,  and 
p  the  pressure  in  pounds  per 
square  unit  applied  to  a ;  then 
the  unit-pressure  in  the  fluid 
is  /?,  and  the  total  pressure  on 
the  small  piston  is  pa,  while  that  on  the  large  piston  is  pA. 
Let  the  distances  through  which  the  pistons  move  during 
one.  stroke  be  d  and  D.  Then  the  imparted  work  is  pad, 
and  the  performed  work,  neglecting  frictional  resistances, 
is  pAD.  Consequently  ad=AD,  and  since  a  is  small  as 
compared  with  A,  the  distance  D  must  be  small  compared 
with  d.  Here  is  found  an  illustration  of  the  popular  maxim 
'"  What  is  gained  in  force  is  lost  in  velocity." 

The  Keely  motor,  one  of  the  delusions  of  the  nineteenth 
century,  is  said  to  have  employed  this  principle  to  produce 
some  of  its  effects.  Very  small  pipes,  supposed  by  the 
spectators  to  be  wires  conveying  some  mysterious  force, 
were  used  to  transmit  the  pressure  of  water  to  a  receiver 
where  the  total  pressure  became  very  great  in  consequence 
of  greater  area. 

In  consequence  of  its  fluidity  the  pressure  existing  at 
any  point  in  a  body  of  water  is  exerted  in  all  directions 
with  equal  intensity.  If  water  be  confined  by  a  bounding 
surface,  as  in  a  vessel,  its  pressure  against  that  surface 
must  be  normal  at  every  point,  for  if  it  were  inclined  the 
water  would  move  along  the  surface.  When  water  has  a 
free  surface  the  unit-pressure  at  any  depth  depends  only 
on  that  depth  and  not  on  the  shape  of  the  vessel.  Thus 
in  the  second  diagram  of  Fig.  10a  the  unit-pressure  at  C 


24  HYDROSTATICS  CHAP.  II 

produced  by  the  smaller  column  of  water  aC  is  the  same 
as  that  caused  by  the  larger  column  AC,  and  the  total 
pressure  on  the  base  B  is  the  product  of  its  area  into  the 
unit-pressure  caused  by  the  depth  AB. 

Prob.  10.  In  a  hydrostatic  press  the  diameter  of  the  small 
piston  is  2  inches  and  that  of  the  large  piston  is  12  inches, 
while  the  pressure  in  the  fluid  is  1750  pounds  per  square  inch. 
If  the  small  piston  moves  3  inches  per  minute,  how  far  does  the 
large  one  move?  What  horse-power  is  transmitted  by  the  press  ? 

ART.  11.     HEAD  AND  PRESSURE 

The  free  surface  of  water  at  rest  is  perpendicular  to 
the  direction  of  the  force  of  gravity,  and  for  bodies  of  water 
of  small  extent  this  surface  may  be  regarded  as  a  plane. 
Any  depth  below  this  plane  is  called  a  "  head,"  or  the  head 
upon  any  point  is  its  vertical  depth  below  the  level  surface. 
In  Art.  10  it  was  seen  that  the  unit  -pressure  at  any  depth 
depends  only  on  the  head  and  not  on  the  shape  of  the 
vessel.  Let  h  be  the  head  and  w  the  weight  of  a  cubic  unit 
of  water;  then  at  the  depth  h  one  horizontal  square  unit 
bears'  a  pressure  equal  to  the  weight  of  a  column  of  water 
whose  height  is  h,  and  whose  cross-section  is  one  square 
unit,  or  wh.  But  the  pressure  at  this  point  is  exerted  in 
all  directions  with  equal  intensity.  The  unit-pressure  p 
at  the  depth  h  then  is  wh,  and  conversely  the  depth,  or 
head,  for  a  unit-pressure  p  is  p/w,  or 

p=wh  h=p/w 


If  h  be  expressed  in  feet  and  p  in  pounds  per  square  foot, 
these  formulas  become,  using  the  mean  value  of  w, 


Thus  pressure  and  head  are  mutually  convertible,  and  in 
fact  one  is  often  used  as  synonymous  with  the  other, 
although  really  each  is  proportional  to  the  other.  Any 


ART.  11  HEAD  AND  PRESSURE  25 

unit-pressure  p  can  be  regarded  as  produced  by  a  head  h, 
which  is  frequently  called  the  "  pressure  head." 

In  engineering  work  p  is  usually  taken  in  pounds  per 
square  inch,  while  h  is  expressed  in  feet.  Thus  the  pressure 
in  pounds  per  square  foot  is  62 . 5/1,  and  the  pressure  in  pounds 
per  square  inch  is  -^  of  this,  or 

£=0.4340^  h  =  2.$o4p  (11), 

These  rules  may  be  stated  in  words  as  follows : 

i  foot  head  corresponds  to  0.434  pounds  per  square  inch; 
i  pound  per  square  inch  corresponds  to  2.304  feet  head. 

These  values,  be  it  remembered,  depend  upon  the  assump- 
tion that  62.5  pounds  is  the  weight  of  a  cubic  foot  of  water, 
and  hence  are  liable  to  variation  in  the  third  significant 
figure  (Art.  4).  The  extent  of  these  variations  for  fresh 
water  may  be  seen  in  Table  13,  which  gives  multiples  of  the 
above  values,  and  also  the  corresponding  quantities  when 
the  cubic  foot  is  taken  as  62.3  pounds. 

The  atmospheric  pressure,  whose  average  value  is  14.7 
pounds  per  square  inch,  is  transmitted  through  water,  and 
is  to  be  added  to  the  pressure  due  to  the  head  whenever  it 
is  necessary  to  regard  the  absolute  pressure.  This  is  im- 
portant in  some  investigations  on  the  pumping  of  water, 
and  in  a  few  other  cases  where  a  partial  or  complete  vacuum 
is  produced  on  one  side  of  a  body  of  water.  For  example, 
if  the  air  be  exhausted  from  a  small  globe,  so  that  its 
tension  is  only  6.5  pounds  per  square  inch,  and  it  be  sub- 
merged in  water  to  a  depth  of  250  feet,  the  absolute  pressure 
per  square  inch  on  the  globe  is 

£=0.434X250  +  14.7  =  123.2  pounds 
and  the  resultant  effective  pressure  per  square  inch  is 
p'  =  123.2  —  6.5  =  116.7  pounds. 

Unless  otherwise  stated,  however,  the  atmospheric  pressure 
need  not  be  regarded,  since  under  ordinary  conditions  it 


26  HYDROSTATICS  CHAP,  n 

acts  with  equal  intensity  upon  both  sides  of  a  submerged 
surface. 

Prob.  lla.  How  many  feet  head  correspond  to  a  pressure 
of  100  pounds  per  square  inch?  How  many  pounds  per  square 
inch  correspond  to  a  head  of  230  feet? 

Prob.  lib.  In  the  first  diagram  of  Fig.  10a  the  diameter 
of  AC  is  }  inches  and  that  of  BC  is  12  inches.  Compute  the 
total  pressure  on  the  base  if  the  heights  AC  and  CB  be  3  and 
2  feet. 

ART.  12.     Loss  OF  WEIGHT  IN  WATER 

It  is  a  familiar  fact  that  bodies  submerged  in  water  lose 
part  of  their  weight ;  a  man  can  carry  under  water  a  large 
stone  which  would  be  difficult  to  lift  in  air  and  timber 
when  submerged  has  a  negative  weight  or  tends  to  rise 
to  the  surface.  The  following  is  the  law  of  loss  which  was 
discovered  by  Archimedes,  about  250  B.C.,  when  considering 
the  problem  of  King  Hiero's  crown: 

The  weight  of  a  body  submerged  in  water  is  less  than 
its  weight  in  air  by  the  weight  of  a  volume  of  water  equal 

to  that  of  the  body. 

i , 

To  demonstrate  this,  consider  that  the  submerged  body 
is  acted  upon  by  the  water  pressure  in  all  directions,  and 
that  the  horizontal  components  of  these  pressures  must 
balance.  Any  vertical  elementary  prism  is  subjected  to 
an  upward  pressure  upon  its  base  which  is  greater  than  the 
downward  pressure  upon  its  top,  since  these  pressures  are 
due  to  the  heads.  Let  hl  be  the  head  on  the  top  of  the 

elementary  prism  and  h2  that  on  its  base, 

land  a  the  cross-section  of  the  prism; 
then  the  downward  pressure  is  wah^  and 
the  upward  pressure  is  wahy  The  differ- 
ence of  these,  wa(h2  —  h^)  is  the  resultant 
upward  water  pressure,  and  this  is  equal 
to  the  weight  of  a  column  of  water  whose 
cross-section  is  a  and  whose  height  is  that  of  the  elementary 


ART.  13  DEPTH  OF  FLOTATION  27 

prism.  Extending  this  theorem  to  all  the  elementary 
prisms,  it  is  concluded  that  the  weight  of  the  body  in  water 
is  less  than  its  weight  in  air  by  the  weight  of  an  equal 
volume  of  watgr. 

It  is  important  to  regard  this  loss  of  weight  in  construc- 
tions Binder  water.  If,  for  example,  a  dam  of  loose  stones 
allows  the  water  to  percolate  through  it,  its  weight  per 
cubic  foot  is  less  than  its  weight  in  air,  so  that  it  can  be 
more  easily  moved  by  horizontal  forces.  As  stone  weighs 
about  150  pounds  per  cubic  foot  in  air,  its  weight  in  water 
is  only  about  150  —  62=88  pounds  per  cubic  foot.  If  a 
cubic  foot  of  sand,  having  voids  amounting  to  40  percent 
of  its  volume,  weighs  no  pounds,  its  loss  of  weight  in  water 
is  0.60X62.5=37.5  pounds,  so  that  its  weight  in  water  is 
110  —  37.5  =72.5  pounds. 

The  ratio  of  the  weight  of  a  substance  to  that  of-  an 
equal  volume  of  water  is  called  the  specific  gravity  of  the 
substance,  and  this  is  easily  computed  from  the  law  of 
Archimedes  after  weighing  a  piece  of  it  in  air  and  then  in 
water ;  or,  if  w  be  the  weight  of  a  cubic  unit  of  water  and 
w'  the  weight  of  a  cubic  unit  of  any  substance,  the  ratio 
w' '/w  is  the  specific  gravity  of  the  substance. 

Prob.  12.  A  box  containing  1.17  cubic  feet  weighs  19.3 
pounds  when  empty  and  133.5  when  filled  with  sand.  It  is 
then  found  that  29.7  pounds  of  water  can  be  poured  in  before 
overflow  occurs.  Show  that  the  percentage  of  voids  in  the 
sand  is  40.6,  that  the  specific  gravity  of  the  sand  mass  is  1.56, 
and  that  the  specific  gravity  of  a  grain  of  sand  is  2.65. 

ART.  13.     DEPTH  OF  FLOTATION 

When  a  body  floats  upon  water  it  is  sustained  by  an 
upward  pressure  of  the  water  equal  to  its  own  weight,  and 
this  pressure  is  the  same  as  the  weight  of  the  volume  of 
water  displaced  by  the  body.  Let  W  be  the  weight  of 
the  floating  body  in  air,  and  W  be  the  weight  of  the  dis- 


28 


HYDROSTATICS 


CHAP.  II 


placed  water ;  then  W  =  W.  Now  let  z  be  the  depth  of 
flotation  of  the  body;  then  to  find  its  value  for  any  par- 
ticular case  W  is  to  be  expressed  in  terms  of  the  linear 
dimensions  of  the  body,  and  W  in  terms  of  the  depth  of 
flotation  z. 

For  example,  a  cone  which  weighs  w'  pounds  per  cubic 
foot  floats  with  its  base  downward,  its  altitude  being  d 

and  the  radius  of  its  base  b.  The 
weight  of  the  floating  cone  is 

W  =wr  .7tb\\d 

and  the  weight  of  the  displaced 
water  is  that  of  a  frustum  of  the 
altitude  z,  or 


FIG.  13a 


3          \   a     I       3 
Equating  these  values  and  solving  for  z  gives 


which  is  the  depth  of  flotation.  '  Here  w' /w  is  the  specific 
gravity  of  the  floating  body. 

To  find  the  depth  of  flotation  for  a  cylinder  lying  hori- 
zontally, let  w'  be  its  weight  per  cubic  unit,  /  its  length, 
and  r  the  radius  of  its  cross- 
section.  The  depth  of  flota- 
tion is  DE,  or  if  0  be  the 
angle  ACE, 

z  =  (i—cos6)r 

The  weight  of  the  cylinder  is 

W'=7tr2l.wf  FIG.  136 

and  that  of  the  displaced  water  is 

r2  sin0  cosd^l.w 


ART.  14  STABILITY  OF  FLOTATION  29 

Equating  the  values  of  W  and  W ',  and  substituting  for 
sin 6  cos#  its  equivalent  J  sin20,  there  results 

2  arc#  —  sin20  =  27rs> 

in  which  s  represents  the  ratio  w'/w  or  the  specific  gravity 
of  the  material  of  the  cylinder.  From  this  equation  0 
is  to  be  found  by  trial  for  any  particular  case,  and  then  z 
is  computed.  For  example,  if  ^'=26.5  pounds  per  cubic 
foot,  then  5-  is  0.424,  and 

2  arc#  — sin2$  —  2.664  =  0 

To  solve  this  equation,  values  are  to  be  assumed  for  6, 
until  one  is  found  that  satisfies  it ;  thus  from  Table  52, 

for  6  =  83°        2.897  —  0.242  —  2.664  =  —0.009 
for  #=83^        2.906  —  0.234  —  2.664=  +0.008 

Therefore  6  lies  between  83°  and  83°  15',  and  is  probably 
about  83°  8'.  Hence  the  depth  of  flotation  is  z  =  (i  —  o.i2o)r 
=  o.88r,  or  if  the  diameter  be  one  foot  the  depth  of  flota- 
tion is  0.44  feet. 

Prob.  13a.  Show  that  the  depth  of  flotation  for  a  sphere 
of  radius  r  is  one  of  the  positive  roots  of  the  cubic  equation 
z*  —  T,rz2  +  4r3s  =  o.  If  the  diameter  of  a  sphere  is  2  feet  and 
its  specific  gravity  0.65,  find  the  depth  z. 

Prob.  136.  A  wooden  stick  i\  inches  square  and  10  feet  long 
is  to  be  used  for  a  velocity  float  which  is  to  stand  vertically  in 
the  water.  How  many  square  inches  of  sheet  lead  ^  inch 
thick  must  be  tacked  on  the  sides  of  this  stick  so  that  only  4 
inches  will  project  above  the  water  surface?  The  wood  weighs 
27  and  the  lead  710  pounds  per  cubic  foot. 

ART.  14.     STABILITY  OF  FLOTATION 

The  equilibrium  of  a  floating  body  is  stable  when  it 
returns  to  its  primitive  position  after  having  been  slightly 
moved  therefrom  by  extraneous  forces,  it  is  indifferent 
when  it  floats  in  any  position,  and  it  is  unstable  when  the 


30 


HYDROSTATICS 


CHAP.  II 


FIG.  14 


slightest  force  causes  it  to  leave  its  position  of  flotation. 
For  instance,  a  short  cylinder  with  its  axis  vertical  floats 
in  stable  equilibrium,  but  a  long  cylinder  in  this  position 
is  unstable,  and  a  slight  force  causes  it  to  fall  over  and  float 
with  its  axis  horizontal  in  indifferent  equilibrium.  It  is 
evident  that  the  equilibrium  is  the  more  stable  the  lower 
the  center  of  gravity  of  the  body. 

The  stability  depends  in  any  case  upon  the  relative 
position  of  the  center  of  gravity  of  the  body  and  its  center 
of  buoyancy,  the  latter  being  the  center  of  gravity  of  the 

displaced  water.  Thus  in  Fig. 
14  let  G  be  the  center  of  gravity 
of  the  body  and  let  C  be  its 
center  of  buoyancy  when  in  an 
upright  position.  Now  if  an  ex- 
traneous force  causes  the  body 
to  tip  into  the  position  shown  t 
the  center  of  gravity  remains 
at  G,  but  the  center  of  buoyancy  moves  to  D.  In  this  new 
position  of  the  body  it  is  acted  upon  by  the  forces  W  and 
W,  which  are  equal  and  parallel  but  opposite  in  direction. 
These  forces  form  a  couple  which  tends  either  to  restore 
the  body  to  the  upright  position  or  to  cause  it  to  deviate 
farther  from  that  position.  Let  the  vertical  through  D 
be  produced  to  meet  the  center  line  CG  in  M.  If  M  is 
above  G  the  equilibrium  is  stable,  as  the  forces  W  and  W 
tend  to  restore  it  to  its  primitive  position;  if  M  coincides 
with  G  the  equilibrium  is  indifferent;  and  if  M  be  below 
G  the  equilibrium  is  unstable. 

The  point  M  is  called  the  '  metacenter, '  and  the  theorem 
may  be  stated  that  the  equilibrium  is  stable,  indifferent,  or 
unstable  according  as  the  metacenter  is  above,  coincident 
with,  or  below  the  center  of  gravity  of  the  body.  The 
measure  of  the  stability  of  a  stable  floating  body  is  the 
moment  of  the  couple  formed  by  the  forces  W  and  W. 


ART.  15 


NORMAL  PRESSURE 


31 


But  GM  is  proportional  to  the  lever  arm  of  the  couple,  and 
hence  the  quantity  W  X  GM  may  be  taken  as  a  measure  of 
stability.  The  stability,  therefore,  increases  with  the  weight 
of  the  body,  and  with  the  distance  of  the  metacenter  above 
the  center  of  gravity.  (See  Art.  180.) 

The  most  important  application  of  these  principles  is 
in  the  design  of  ships,  and  usually  the  problems  are  of  a 
complex  character  which  can  only  be  solved  by  tentative 
methods.  The  rolling  of  the  ship  due  to  lateral  wave  action 
must  also  receive  attention,  and  for  this  reason  the  center 
of  gravity  should  not  be  put  too  low. 

Prob.  14.  A  square  prism  of  uniform  specific  gravity  s 
has  the  length  h  and  the  cross-section  b2.  When  placed  in 
water  with  its  axis  vertical,  show  that  it  is  in  stable,  indifferent, 
or  unstable  equilibrium  according  as  b2  is  greater,  equal  to, 
or  less  than  6h2s(i—  s).  When  placed  with  its  axis  at  an  incli- 
nation of  45°,  show  that  it  will  assume  the  vertical  or  horizontal 
position  according  as  b2  is  greater  or  less  than  4h2s(i—  s). 

ART.  15.     NORMAL  PRESSURE 

The  total  normal  pressure  on  any  immersed  surface  may 
be  found  by  the  following  theorem: 

The  total  normal  pressure  is  equal  to  the  product  of  the 
weight  of  a  cubic  unit  of  water,  the  area  of  the  surface, 
and  the  head  on  its  center  of  gravity. 

To  prove  this  let  A  be  the  area  of  the  surface,  and  imagine 
it  to  be  composed  of  elementary  areas,  av  a2,  a3,  etc.,  each 
of  which  is  so  small  that 
the   unit-pressure   over   it 
may  be  taken  as  uniform ; 
let  kv  h2,  /z3,  etc.,  be  the 
heads  on  these  elementary 
areas,    and    let  w   denote 
the  weight  of  a  cubic  unit 
of  water.     The  unit -pressures  at  the  depths  hv  h2,  h3,  etc., 


32  HYDROSTATICS  CHAP,  n 

are  whv  wh2,  wh3,  etc.  (Art.  11),  and  hence  the  normal 
pressures  on  the  elementary  areas  av  a2,  aB,  etc.,  are  wajiv 
wa2h2,  wa3h3,  etc.  The  total  normal  pressure  P  on  the 
entire  surface  then  is 

P  =w(a1hl  +  a2h2  +  a3h3  +  etc.) 

Now  let  h  be  the  head  on  the  center  of  gravity  of  the  sur- 
face ;  then,  from  the  definition  of  the  center  of  grayity, 


as^3  +  etc.  =  Ah 
Therefore  the  normal  pressure  is 

'p=wAh  (15) 

which  proves  the  theorem  as  stated. 

This  rule  applies  to  all  surfaces,  whether  plane,  curved, 
or  warped,  and  however  they  be  situated  with  reference  to 
the  water  surface.  Thus  the  total  normal  pressure  upon 
the  surface  of  an  immersed  cylinder  remains  the  same  what- 
ever be  its  position,  provided  the  depth  of  the  center  of 
gravity  of  that  surface  be  kept  constant.  It  is  best  to  take 
h  in  feet,  A  in  square  feet,  and  w  as  62.5  pounds  per  cubic 
foot  ;  then  P  will  be  in  pounds.  In  case  surfaces  are  given 
whose  centers  of  gravity  are  difficult  to  determine,  they 
should  be  divided  into  simpler  surfaces,  and  then  the  total 
normal  pressure  is  the  sum  of  the  normal  pressures  on  the 
separate  surfaces. 

The  normal  pressure  on  the  base  of  a  vessel  filled  with 
water  is  equal  to  the  weight  of  a  cylinder  of  water  whose 
base  is  the  base  of  the  vessel,  and  whose  height  is  the  depth 
of  water.  Only  in  the  case  of  a  vertical  cylinder  does  this 
become  equal  to  the  weight  of  the  water,  for  the  pressure 
on  the  base  of  a  vessel  depends  upon  the  depth  of  water  and 
not  upon  the  shape  of  the  vessel.  Also  in  the  case  of  a  dam, 
the  depth  of  the  water  and  not  the  size  of  the  pond  deter- 
mines the  amount  of  pressure. 


ART.  16  PRESSURE  IN  A  GIVEN  DIRECTION  33 

When  a  surface  is  plane  the  total  normal  pressure  is 
the  resultant  of  all  the  parallel  pressures  acting  upon  it. 
This  is  not  true  for  curved  surfaces,  for,  as  the  pressures 
have  different  directions,  their  resultant  is  not  equal  to 
their  numerical  sum,  but  must  be  obtained  by  the  rules  for 
the  composition  of  forces.  For  example,  if  a  sphere  of 
diameter  d  be  filled  with  water  the  total  normal  pressure  as 
found  by  the  formula  (15)  is 


"but  the  resultant  pressure  is  nothing,  for  the  elementary 
normal  pressures  act  in  all  directions  so  that  no  tendency 
to  motion  exists.  The  weight  of  water  in  this  sphere  is 
-J-WTrd3,  or  one  third  of  the  total  normal  pressure,  and  the 
direction  of  this  is  vertical. 

Prob.  15a.  A  board  3  feet  wide  at  one  end  and  2.5  feet  wide 
at  the  other  end  is  8  feet  long.  What  is  the  normal  pressure 
upon  each  of  its  sides  when  placed  vertically  in  water  with  the 
narrow  end  in  the  surface? 

Prob.  156.  An  ellipse,  with  major  and  minor  axes  equal  to 
12  and  8  feet,  is  submerged  so  that  one  extremity  of  the  major 
axis  is  3.5  and  the  other  8.5  feet  below  the  water  surface.  Show 
that  the  normal  pressure  on  one  side  is  28  300  pounds. 


ART.  16.     PRESSURE  IN  A  GIVEN  DIRECTION 

The  pressure  against  an  immersed  plane  surface  in  a 
given  direction  may  be  found  by  obtaining  the  normal 
pressure  by  Art.  15  and  computing  its  component  in  the 
required  direction,  or  by  means  of  the  following  theorem : 

The  horizontal  pressure  on  any  plane  surface  is  equal 
to  the  normal  pressure  on  its  vertical  projection;  the 
vertical  pressure  is  equal  to  the  normal  pressure  on  its 
horizontal  projection;  and  the  pressure  in  any  direction  is 
equal  to  the  normal  pressure  on  a  projection  perpendicular 
to  that  direction. 


34 


HYDROSTATICS 


CHAP.  II 


To  prove  this  let  A  be  the  area  of  the  given  surface,  repre- 

sented by  AA  in  Fig.  16a,  and  P 
the  normal  pressure  upon  it,  or 
P  =wAh.  Now  let  it  be  required 
to  find  the  pressure  P'  in  a  direction 
making  an  angle  0  with  the  nor- 
mal to  the  given  plane.  Draw 
ArAf  perpendicular  to  the  direc- 
tion of  P'  ,  and  let  A1  be  the  area 

of  the  projection  of  A  upon  it.     The  value  of  P'  then  is 

P'=P  co$d=wAhcosd 
But  A  cos#  is  the  value  of  A'  by  the  construction.     Hence 


FIG.  16a 


P'=wA'k 
and  the  theorem  is  thus  demonstrated. 


(16) 


This  theorem  does  not  in  general  apply  to  curved  sur- 
faces. But  in  cases  where  the  head  of  water  is  so  great  that 
the  pressure  may  be  regarded 
as  uniform  it  is  also  true  for 
curved  surfaces.  For  instance, 
consider  a  cylinder  or  sphere 
subjected  on  every  elementary 
area  to  the  unit-pressure  p  due 
to  the  high  head  h,  and  let  it 
be  required  to  find  the  pressure 
in  the  direction  shown  by  qv  q2, 
and  <?3  in  Fig.  166.  The  pres- 
sures pv  p2,  p3,  etc.,  on  the  elementary  areas  av  a2,  a3,  etc., 
have  the  values 


FIG.  166 


and  the  components  of  these  in  the  given  direction  are 

,  etc., 


ART.  17         CENTER  OF  PRESSURE  ON  RECTANGLES  35 

whence  the  total  pressure  Pf  in  the  given  direction  is 
Pr  =p(al  cosd1  +  a2  cos#2  +  a3  cos#3+etc.) 

But  the  quantity  in  the  parentheses  is  the  projection  of 
the  given  surface  upon  a  plane  perpendicular  to  the  given 
direction,  or  MN.  Hence  there  results 


which  is  the  same  rule  as  for  plane  surfaces. 

For  the  case  of  a  water  pipe  let  p  be  the  interior  pressure 
per  square  inch,  /  its  thickness,  and  d  its  diameter  in  inches. 
Then  for  a  length  of  one  inch  the  force  tending  to  rupture 
the  pipe  longitudinally  is  pd.  The  tensile  unit-stress  5 
in  the  walls  of  the  pipe  acting  over  the  area  2t  constitutes 
the  resisting  force  2/5.  As  these  forces  are  equal,  it  follows 
that  2St=--pd  is  the  fundamental  equation  for  the  discus- 
sion of  the  strength  of  water  pipes  under  static  water 
pressure.  For  example,  if  the  tensile  strength  of  cast  iron 
be  20  ooo  pounds  per  square  inch,  the  pressure  p  required 
to  burst  a  pipe  24  inches  in  diameter  and  0.75  inches  thick 
is  1250  pounds  per  square  inch,  which  corresponds  to  a 
head  of  2880  feet. 

Prob.  16a.  A  circular  plate  3  feet  in  diameter  is  immersed 
so  that  the  head  on  its  center  is  18  feet,  its  plane  making  an 
angle  of  27°  with  the  vertical.  Compute  the  horizontal  and 
vertical  pressures  upon  one  side  of  it. 

Prob.  166.  What  should  be  the  thickness  of  a  water  pipe 
1  8  inches  in  diameter  in  order  that  the  tensile  unit-stress  in 
it  may  be  1600  pounds  per  square  inch  when  under  a  head  of 
water  of  230  feet? 

ART.  17.     CENTER  OF  PRESSURE  ON  RECTANGLES 

The  center  of  pressure  on  a  surface  immersed  in  water 
is  the  point  of  application  of  the  resultant  of  all  the  nor- 
mal pressures  upon  it.  The  simplest  case  is  the  following: 


36 


HYDROSTATICS 


CHAP.  II 


If  a  rectangle  be  placed  with  one  end  in  the  water  sur- 
face, the  center  of  pressure  is  distant  from  that  end  two 
thirds  of  its  length. 

This  theorem  will  be  proved  by  the  help  of  the  graph- 
ical illustration  shown  in  Fig.  17  a.      The  rectangle,  which 

in  practice  might  be  a  board, 
is  placed  with  its  breadth  per- 
pendicular to  the  plane  of  the 
drawing,  so  that  AB  repre- 
sents its  edge.  It  is  required 
to  find  the  center  of  pressure 
C.  For  any  head  h  the  unit- 
pressure  is  wh  (Art.  15),  and 
hence  the  unit-pressures  on 
one  side  of  A  B  may  be  graphically  represented  by  arrows 
which  form  a  triangle.  Now  if  a  force  P  equal  to  the  total 
pressure  is  applied  on  the  other  side  of  the  rectangle  to 
balance  these  unit-pressures,  it  must  be  placed  opposite 
to  the  center  of  gravity  of  the  triangle.  Therefore  AC 
equals  two  thirds  of  AB,  and  the  rule  is  proved.  The 
head  on  C  is  evidently  also  two  thirds  of  the  head  on  B. 


FIG.  17  a 


Another  case  is  that  shown  in  Fig.  17  b,  where  the  rect- 
angle, whose  length  is  B±B21  is  wholly  immersed,  the  head 
on  Bl  being  hv  and  on  B2       A 
being    hy     Let    ABl=b19 
AC=y,andAB2  =  b2.  Now 
the  normal  pressure  Pl  on 


>x  is  applied  at  the  dis- 
tance |6X  from  A,  and  the 
normal  pressure  P2  on 

AB2  is  applied  at  the  distance  |62  from  A.  The  normal 
pressure  P  on  BtB2  is  the  difference  of  Pl  and  P2,  or  P  = 
P2  — P1.  Also  by  taking  moments  about  A  as  an  axis, 


ART.  17        CENTER  OF  PRESSURE  ON  RECTANGLES  37 

Now,  by  Art.  15,  the  normal  pressures  P2  and  P1  for  a  rect- 
angle one  unit  in  breadth  are  P2  =  %wb2h2  and  Pv  =  %wbji^ 
whence  the  total  normal  pressure  is  P  =  %w(b2h2  —  blh1),  and 
accordingly  the  center  of  pressure  is  given  by 

.v*.-v*. 

* 


Now  if  6  be  the  angle  of  inclination  of  the  plane  to  the 
water  surface  the  values  of  h2  and  hi  are  b2  sin#  and  b^  sin#. 
Accordingly  the  expression  becomes 


(17)« 


Again,  if  hf  be  the  head  on  the  center  of  pressure,  y  = 
hf  cosectf,  b2=h2cosec6,  and  bl=hlcosecO.  These  inserted 
in  the  last  equation  give 

h,  'V-v  fl7) 

3*iJ-V  (    '' 

These  formulas  are  very  convenient  for  computation,  as 
the  squares  and  cubes  may  be  taken  from  tables. 

If  /^  equals  h2  the  above  formula  becomes  indeterminate, 
which  is  due  to  the  existence  of  the  common  factor  h2  —  hl 
in  both  numerator  and  denominator  of  the  fraction  ;  divid- 
ing out  this  common  factor,  it  becomes 

2     V  +  ft.ft.+V          \ 

~3~      *,  +  *, 
from  which,  if  h2  =hl  =h,  there  is  found  the  result  h'  =h. 

Prob.  17a.  In  Fig.  17a  let  the  length  of  AB  be  8  feet  and 
its  inclination  to  the  vertical  be  30  degrees.  Find  the  depth 
of  the  center  of  pressure. 

Prob.  176.  A  rectangle  8  feet  long  is  immersed  in  water  with 
its  ends  parallel  to  the  surface,  the  head  on  one  end  being  7  feet 
and  that  on  the  other  5  feet.  Find  the  head  on  the  center  of 
pressure,  and  also  the  value  of  P. 


38  HYDROSTATICS  CHAP,  n 


ART.  18.     GENERAL  RULE  FOR  CENTER  OF  PRESSURE 

For  any  plane  surface  immersed  in  a  liquid,  the  center 
of  pressure  may  be  found  by  the  following  rule  : 

Find  the  moment  of  inertia  of  the  surface  and  its  stati- 
cal moment,  both  with  reference  to  an  axis  situated  at  the 
intersection  of  the  plane  of  the  surface  with  the  water 
level.  Divide  the  former  by  the  latter,  and  the  quotient 
is  the  perpendicular  distance  from  that  axis  to  the  center 
of  pressure. 

The  demonstration  is  analogous  to  that  in  the  last 
article.  Let  BJ3.2  in  Fig.  176  be  the  trace  of  the  plane  sur- 
face, which  itself  is  perpendicular  to  the  plane  of  the  draw- 
ing, and  C  be  the  center  of  pressure,  at  a  distance  y  from 
A  where  the  plane  of  the  surface  intersects  the  water  level. 
Let  alt  a2,  a3,  etc.,  be  elementary  areas  of  the  surface,  and 
^i>  ^2»  ^s»  e"tc->  "the  heads  upon  them,  which  produce  the 
normal  elementary  pressures,  waji^  wa2h2,  wa3h3l  etc.  Let 
y»  ?2>  ?3>  etc.,  be  the  distances  from  A  to  these  elementary 
areas.  Then  taking  the  point  A  as  a  center  of  moments, 
the  definition  of  center  of  pressure  gives  the  equation 

(walh1  +  wa2h2  +  wa3h3  +  etc.)?  = 

1  +  wa2h2y2  +  wa3h3y3  +  etc. 


Now  let  6  be  the  angle  of  inclination  of  the  surface  to  the 
water  level;  then  h1=ylsin6,  h2=y2smO,  h3=y3sin6,  etc. 
Hence,  inserting  these  values,  the  expression  for  y  is 

_  o^2  +  a2y22  +  a3y32  +  etc. 


The  numerator  of  this  fraction  is  the  sum  of  the  products 
obtained  by  multiplying  each  element  of  the  surface  by 
the  square  of  its  distance  from  the  axis,  which  is  called  the 
moment  of  inertia  of  the  surface.  The  denominator  is  the 
sum  of  the  products  obtained  by  multiplying  each  element 


ART.  is      GENERAL  RULE  FOR  CENTER  OF  PRESSURE         39 

of  the  surface  by  its  distance  from  the  axis,  which  is  called 
the  statical  moment  of  the  surface.  Therefore 

moment  of  inertia  _  /' 
statical  moment    =  S~ 

is  the  general  rule  for  finding  the  position  of  the  center  of 
pressure  of  an  immersed  plane  surface. 

The  statical  moment  of  a  surface  is  simply  its  area  mul- 
tiplied by  the  distance  of  its  center  of  gravity  from  the  given 
axis.  The  moments  of  inertia  of  plane  surfaces  with  refer- 
ence to  an  axis  through  the  center  of  gravity  are  deduced 
in  works  on  theoretical  mechanics;  the  following  are  a 
few  values,  the  axis  being  parallel  to  the  base  of  the  rect- 
angle or  triangle: 

for  a  rectangle  of  base  b  and  depth  d,     I 

for  a  triangle  of  base  b  and  altitude  d,    I  = 

for  a  circle  with  diameter  d,  /=-g-1¥7rd4 

To  find  from  these  the  moment  of  inertia  with  reference  to 
a  parallel  axis,  the  well-known  formula  I'=I  +  Ak2  is  to 
be  used,  where  A  is  the  area  of  the  surface,  k  the  distance 
from  the  given  axis  to  the  center  of  gravity  of  the  sur- 
face, and  /'  the  moment  of  inertia  required. 

For  example,  let  it  be  required  to  find  the  center  of 
pressure  of  a  vertical  circle  immersed  so  that  the  head  on 
its  center  is  equal  to  its  radius.  The  area  of  the  circle  is 
^7r<i2,  and  its  statical  moment  with  reference  to  the  upper 
edge  is  \nd2X±d.  Then  from  (4) 


J*d'.# 

or  the  center  of  pressure  is  at  a  distance  J d  below  the  cen- 
ter of  the  circle. 

Prob.  18.  Find  the  center  of  pressure  for  the  triangle  in 
Fig.  13a  when  its  vertex  is  in  the  water  surface.  Also  the  center 
of  pressure  when  the  base  is  in  the  surface. 


40 


HYDROSTATICS 


CHAP.  It 


ART.  19.     PRESSURE  ON  GATES  AND  DAMS 

In  the  case  of  an  immersed  plane  the  water  presses 
equally  upon  both   sides   so  that  no   disturbance  of  the 

equilibrium  results  from  the  pres- 
sure. But  in  case  the  water  is 
at  different  levels  on.  opposite  sides 
of  the  surface  the  opposing  pres- 
sures are  unequal.  For  example,  the 
cross-section  of  a  self-acting  tide- 
=  gate,  built  to  drain  a  salt  marsh  is 
shown  in  the  figure.  On  the  ocean 
side  there  is  a  head  of  hl  above  the 
sill,  which  gives  for  every  linear  foot 
of  the  gate  the  horizontal  pressure 


which  is  applied  at  the  distance  J/^  above  the  sill.  On 
the  other  side  the  head  on  the  sill  is  H21  which  gives  the  hori- 
zontal pressure  P2  =  ^wh22  acting  in  the  opposite  direction 
to  that  of  Pj.  The  resultant  horizontal  pressure  is 


and  if  z  be  the  distance  of  the  point  of  application,  of  P 
above  the  sill,  the  equation  of  moments  is 


from  which  z  can  be  computed.  For  example,  if  h±  be 
7  feet  and  H2  be  4  feet  the  resultant  pressure  on  one  linear 
foot  of  the  gate  is  found  to  be  1031  pounds  and  its  point 
of  application  to  be  2.82  feet  above  the  sill.  The  action 
of  this  gate  in  resisting  the  water  pressure  is  like  that  of  a 
beam  under  its  load,  the  two  points  of  support  being  at 
the  sill  and  the  hinge.  If  h  be  the  height  of  the  gate,  the 
reaction  at  the  hinge  is  Pz/h  and  from  the  above  expres- 
sion for  Pz  it  is  seen  that  this  reaction  has  its  greatest  value 


ART.  19  PRESSURE  ON  GATES  AND  DAMS  41 

when  hi  becomes  equal  to  h  and  H2  is  zero.  In  the  case 
of  the  vertical  gate  of  a  canal  lock,  which  swings  horizon- 
tally like  a  door,  a  similar  problem  arises  and  a  similar 
conclusion  results. 

When  the  water  level  behind  a  masonry  dam  is  lower 
than  its  top,  as  in  Fig.  196,  the  water  pressure  on  the  back 
is  normal  to  the  plane  AB,  and  this  may  be  resolved  into 
horizontal  and  vertical  components.  The  horizontal  com- 
ponent is  the  only  one  usually  necessary  to  be  considered, 
and  this  will  be  called  P,  and  its  distance  above  the  base 
of  the  dam  will  be  called  p.  From  Arts.  16  and  17  the 
values  of  these,  for  one  linear  foot  of  the  dam,  are 


in  which  h  is  the  height  of  the  dam  and  w  the  weight  of  a 
cubic  unit  of  water.  The  horizontal  water  pressure  is  hence 
independent  of  the  slope  of  the  back  of  the  dam.  The 
normal  pressure  on  the  back,  however,  is  %wh2  sec/9,  its 
horizontal  component  being  %wh2  and  its  vertical  com- 
ponent %wh2  tan 0. 

When  the  water  runs  over  the  top  of  the  dam  as  in 
Fig.  19c,  let  h  be  the  height  of  the  dam  and  d  the  depth  of 


FIG. 

water  on  the  crest.     Then  by  Art.  16  the  horizontal  pres- 
sure against  the  back  is 


and  by  Art.  17  the  vertical  distance  of  its  point  of  applica- 
tion above  the  base  BD  is  found  to  be 


42  HYDROSTATICS  CHAP,  n 


If  d  =  o,  these  expressions  become  P  =  %wh2  and  p  =  %h. 
If  d  is  infinite,  the  value  of  p  becomes  £/&,  and  hence  in  no 
case  can  the  pressure  P  be  applied  as  high  as  the  middle 
of  the  height  of  the  dam. 

It  is  not  the  place  here  to  enter  into  the  discussion  of 
the  subject  of  the  design  of  masonry  dams,  but  two  ways 
in  which  they  are  liable  to  fail  may  be  noted.  The  first 
is  that  of  sliding  along  a  horizontal  joint,  as  BD  ;  here  the 
horizontal  component  of  the  thrust  overcomes  the  resist- 
ing force  of  friction  acting  along  the  joint.  If  W  be  the 
weight  of  masonry  above  the  joint,  and  /  the  coefficient 
of  friction,  the  resisting  friction  is  fW,  and  the  dam  will 
slide  if  the  horizontal  component  of  the  pressure  is  equal 
to  or  greater  than  this.  The  condition  for  failure  by  slid- 
ing then  is  P=fW.  For  example,  consider  a  masonry 
dam  of  rectangular  cross-section  which  is  4  feet  wide  and 
h  feet  high,  the  water  being  level  with  its  top.  Let  its 
weight  per  cubic  foot  be  140  pounds,  and  let  it  be  required 
to  find  the  height  h  for  which  it  would  fail  by  sliding  along 
the  base,  the  coefficient  of  friction  being  0.70.  The  hori- 
zontal water  pressure  is  JX62.5  Xh2  and  the  resisting  fric- 
tion is  o.yXi4oX4X/£.  Placing  these  equal  there  is  found 
h  =  i2.$  feet. 

The  second  method  of  failure  of  a  masonry  dam  is  by 
overturning,  or  by  rotating  about  the  toe  D.  This  occurs 
when  the  moment  of  P  equals  the  moment  of  W  with 
respect  to  D,  or  if  p  and  q  are  the  lever  arms  dropped  from 
D  upon  the  directions  of  P  and  W  the  condition  for  failure 
by  rotation  is  Pp  =  Wq.  For  example,  if  it  be  required 
to  find  the  height  of  the  above  rectangular  dam  so  that 
it  will  fail  by  rotation,  the  lever  arms  p  and  q  are  %h  and 
2  feet,  and  the  equation  of  moments  with  respect  to  an 
axis  through  the  toe  of  the  dam  is 

i  X  62.  5  xh*X%h  =  140X4X^X2 
from  which  there  is  found  h  =  io.4  feet.     The  horizontal 


ART.  20          HYDROSTATICS  IN  METRIC  MEASURES  43 

water  pressure  for  one  linear  foot  of  the  dam  at  the  instant 
of  failure  is  %wh*  =  33&o  pounds. 


Prob.  19.  A  water  pipe  passing  through  a  masonry  dam  is 
closed  by  a  cast-iron  circular  valve  A  B,  which  is  hinged  at  A, 
and  which  can  be  raised  by  a  vertical 
chain  EC.  The  diameter  of  the 
valve  is  3  feet,  its  plane  makes  an 
angle  of  27°  with  the  vertical,  and 
the  depth  of  its  center  below  the 
water  level  is  12  feet.  Compute  the 
normal  water  pressure  P,  and  the 
distance  of  the  center  of  pressure 
from  the  hinge  A  .  Disregarding  the 

weight  of  the  valve  and  chain,  com- 

.      ,    .  FIG. 

pute  the  force  F  required  to  open 

the  valve.  If  the  weight  of  the  chain  is  25  pounds  and  that 
of  the  valve  240  pounds,  compute  the  force  F. 

ART.  20.     HYDROSTATICS  IN  METRIC  MEASURES 

(Art.  11)  If  the  head  h  be  in  meters  and  the  unit- 
pressure  p  be  in  kilograms  per  square  meter,  the  formulas 
(11)!  become 

p  =  ioooh  h  =  o.ooip 

In  engineering  practice  p  is  usually  taken  in  kilograms 
per  square  centimeter,  while  h  is  expressed  in  meters.  Then 

p  =  o.ih  h  =  iop  (20) 

Stated  in  words  these  practical  rules  are: 

i  meter  head  corresponds  to  o.i  kilogram  per  square 
centimeter  ; 

i  kilogram  per  square  centimeter  corresponds  to  10 
meters  head. 

These  values  depend  upon  the  assumption  that  1000 
kilograms  is  the  weight  of  a  cubic  meter  of  water,  and 
hence  results  derived  from  them  are  liable  to  an  uncertainty 
in  the  third  or  fourth  significant  figure  as  Table  14  shows. 


44  HYDKOSTATICS  CHAP,  n 

The  atmospheric  pressure  of  1.033  kilograms  per  square 
centimeter  is  to  be  added  to  the  pressure  due  to  the  head 
whenever  it  is  necessary  to  regard  the  absolute  pressure. 
For  example,  if  the  air  be  exhausted  from  a  small  globe 
so  that  its  pressure  is  only  0.32  kilograms  per  square 
centimeter  and  it  be  submerged  in  water  to  a  depth  of 
86  meters,  the  absolute  pressure  per  square  centimeter 
on  the  globe  is  0.1X86  +  1.033=9.633  kilograms,  and 
the  resultant  effective  pressure  per  square  centimeter  is 
9.633-0.32=9.313  kilograms. 

(Art.  12)  The  specific  gravity  of  a  substance  is  ex- 
pressed by  the  same  number  as  the  weight  of  a  cubic 
centimeter  in  grams,  or  the  weight  of  a  cubic  decimeter  in 
kilograms,  or  the  weight  of  a  cubic  meter  in  metric  tons. 
Thus,  if  the  specific  gravity  of  stone  is  2.4,  a  cubic  meter 
weighs  2.4  metric  tons  or  2400  kilograms.  A  bar  one 
square  centimeter  in  cross-section  and  one  meter  long 
contains  100  cubic  centimeters;  hence  if  such  a  steel  bar 
be  steel  having  a  specific  gravity  of  7.9,  it  weighs  790 
grams  or  0.79  kilograms  in  air,  while  in  water  it  weighs" 
690  grams  or  0.69  kilograms. 

(Art.  15)  Here  h  is  to  be  taken  in  meters,  A  in  square 
meters,  and  w  as  1000  kilograms  per  cubic  meter;  then 
P  will  be  in  kilograms. 

(Art.  16)  For  a  water  pipe  let  P  be  the  interior  pres- 
sure in  kilograms  per  square  centimeter  and  d  its  diameter 
in  centimeters.  Then  for  a  length  of  one  centimeter  the 
force  tending  to  rupture  the  pipe  longitudinally  is  pd. 
Let  S  be  the  stress  in  kilograms  per  square  centimeter 
in  the  walls  of  the  pipe;  this  acts  over  the  area  2t,  if  t 
be  the  thickness.  As  these  forces  are  equal,  the  equation 
2St=pd  is  to  be  used  for  the  investigation  of  water  pipes. 
For  example,  let  it  be  required  to  find  what  head  will 
burst  a  cast-iron  pipe  60  centimeters  in  diameter  and  2 
centimeters  thick;  the  tensile  strength  of  the  material 


ART.  20  HYDROSTATICS  IN  METRIC  MEASURES  45 

being  1400  kilograms  per  square  centimeter.  Using  the 
equation  the  value  of  p  is  found  to  be  93.3  kilograms  per 
square  centimeter,  and  then,  from  Art.  9,  the  required 
liead  h  is  933  meters. 

(Art.  19)  Consider  a  rectangular  masonry  dam  which 
weighs  2400  kilograms  per  cubic  meter  and  which  is  1.4 
meters  thick.  First,  let  it  be  required  to  find  the  height 
of  water  for  which  it  would  fail  by  sliding,  the  coefficient 
of  friction  being  0.75.  The  horizontal  water  pressure  is 
JXioooX/*2,  and  the  resisting  friction  is  0.7 5  X 2400 X 
I.4X/&;  placing  these  equal  there  is  found  ^  =  5.04  meters. 
Secondly,  to  find  the  height  for  which  failure  will  occur 
by  rotation,  the  equation  of  moments  is 

jXioooX/*2Xj/*  =  2400X1.4X^X0. 75 

from  which  there  is  found  ^  =  3.89  meters.  The  horizontal 
water  pressure  for  one  linear  meter  of  this  dam  is  %wh2= 
7560  kilograms. 

Prob.  20a.  In  a  hydrostatic  press  one  half  of  a  metric  horse- 
power is  applied  to  the  small  piston.  The  diameter  of  the 
large  piston  is  30  centimeters  and  it  moves  2  centimeters  per 
minute.  §how  that  the  pressure  in  the  liquid  is  159  kilograms 
per  square  centimeter. 

Prob.  206.  What  is  the  specific  gravity  of  dry  hydraulic 
cement  of  which  20.6  cubic  centimeters  weigh  63.2  grams? 
If  a  cube  of  stone  12.4  centimeters  on  each  edge  weighs  4.88 
kilograms,  what  is  its  specific  gravity? 

Prob.  20c.  In  Fig.  19a  let  the  head  on  one  side  of  the  gate 
be  2.5  and  on  the  other  side  0.6  meters  above  the  sill.  Find 
the  resultant  pressure  for  one  linear  meter  of  the  gate  and  the 
distance  of  its  point  of  application  above  the  sill. 


46  THEORETICAL  HYDRAULICS  CHAP.  IIL 


CHAPTER  III 
THEORETICAL  HYDRAULICS 

ART.  21.     LAWS  OF  FALLING  BODIES 

Theoretical  Hydraulics  treats  of  the  flow  of  water  when 
unretarded  by  opposing  forces  of  friction.  In  a  perfectly 
smooth  inclined  trough  water  would  flow  with  accelerated 
velocity  and  be  governed  by  the  same  laws  as  those  for 
a  body  sliding  down  an  inclined  plane.  Such  a  flow  is, 
however,  never  found  in  practice,  for  all  surfaces  over 
which  water  moves  are  more  or  less  rough.  Friction 
retards  the  motions  caused  by  gravity  so  that  the  theoretic 
velocities  deduced  in  this  chapter  constitute  limits  which 
cannot  be  exceeded  by  the  actual  velocities.  Many  of  the 
laws  governing  the  free  fall  of  bodies  in  a  vacuum  are  sim- 
ilar to  those  of  both  theoretical  and  practical  hydraulics, 
and  hence  they  will  here  be  briefly  discussed. 

When  a  body  is  at  rest  above  the  surface  of  the  earth 
it  immediately  falls  if  its  support  be  removed.  If  the 
fall  occurs  in  a  vacuum  its  velocity  at  the  end  of  one 
second  is  g  feet,  the  mean  value  of  g  being  32.16,  and  at 
the  end  of  /  seconds  its  velocity  is  V=gt.  The  distance 
passed  through  in  the  time  t  is  the  product  of  the  mean 
velocity  \V  by  the  number  of  seconds,  or  h  =  %gt2.  Elim- 
inating t  from  these  two  equations  gives  the  well-known 
relations  between  h  and  V: 

V  =  V^gh       or       h  =  V2/2g:  (21)  r 

which  show  that  the  velocity  varies  with  the  square  root 
of  the  height  and  that  the  height  varies  as  the  square  of 
the  velocity. 


ART.  21  LAWS  OF  FALLING  BODIES  47 

When  a  falling  body  has  the  initial  velocity  u  at  the 
beginning  of  the  time  t  its  velocity  at  the  end  of  this  time 
is  V  =  u  +  gt  and  the  distance  passed  over  in  that  time  is 
h=ut  +  $gt2.  Eliminating  t  from  these  equations  gives 


or     h  =  (V2-u2)/2g  (21), 

as  the  relations  between  V  and  h  for  this  case.  These 
formulas  are  also  true  whatever  be  the  direction  of  the 
initial  velocity  u. 

When  a  body  of  weight  W  is  at  the  height  h  above 
a  given  horizontal  plane  its  potential  energy  •  with  re- 
spect to  this  plane  is  Wh.  When  it  falls  from  rest  to 
this  plane  the  potential  energy  is  changed  into  the  kinetic 
energy  W  .  V2/2g,  if  no  work  has  been  done  against  frac- 
tional resistance,  and  therefore  V2  =  2gh.  If  it  has  an 
initial  velocity  u  in  any  direction  at  the  height  h  above 
the  plane  its  energy  there  is  partly  potential  and  partly 
kinetic,  the  sum  of  these  being  Wh  +  W  .u2/2g't  on  reaching 
the  plane  it  has  the  kinetic  energy  WV2/2g.  Placing  these 
equal  there  results  V2  =  2gh  +  u2  as  found  above  by  an- 
other method.  In  general,  reasoning  from  the  stand- 
point of  energy  is  more  satisfactory  than  that  in  which 
the  element  of  time  is  employed. 

The  general  case  of  a  body  moving  toward  the  earth 
is  represented  in  Fig.  21.  When  the  body  is  at  A  it  is 
at  a  height  h^  above  a  certain  hori- 
zontal plane  and  has  the  velocity 
v^.  When  it  has  arrived  at  B  its 
height  above  the  plane  is  h2  and  its 
velocity  is  v2.  In  the  first  position 
the  sum  of  its  potential  and  kinetic  —  i  - 
energy  with  respect  to  the  given 
horizontal  plane  is 


48  THEORETICAL  HYDRAULICS  CHAP,  in 

and  in  the  second  position  the  sum  of  these  energies  is 


If  no  energy  has  been  lost  between  the  two  positions,  these 
two  expressions  are  equal,  and  hence 


This  equation  contains  two  heights  and  two  velocities, 
and  if  three  of  these  quantities  be  given  the  fourth  can  be 
found.  Thus  if  vlr  hlt  and  h2  be  given,  the  value  of  v2  is 


where  k^  —  h^  is  the  vertical  height  of  A  above  B.  With 
proper  changes  in  notation  this  expression  reduces  to  (21  )2, 
which  is  for  the  case  where  the  horizontal  plane  passes 
through  B,  and  to  (21)^  which  is  the  case  where  there  is 
no  initial  velocity. 

Prob.  21a.  A  body  is  projected  vertically  upward  with  a 
velocity  of  105  feet  per  second.  What  is  its  velocity  after  it 
has  reached  a  height  of  170  feet  above  the  initial  position? 

Prob.  216.  A  body  enters  a  room  through  the  ceiling  with 
a  velocity  of  250  feet  per  second,  and  in  a  direction  making  an 
angle  of  30°  with  the  vertical.  If  the  height  of  the  room  is 
14  feet,  find  the  velocity  of  the  body  as  it  strikes  the  floor, 
resistances  of  the  air  being  neglected. 

ART.  22.     VELOCITY  OF  FLOW  FROM  ORIFICES 

If  an  orifice  be  opened,  either  in  the  base  or  side  of  a 
vessel  containing  water,  the  water  flows  out  with  a  velocity 
which  is  greater  for  high  heads  than  for  low  heads.  The 
theoretic  velocity  of  flow  is  given  by  the  following  theorem 
established  by  Torricelli  in  1644  :  * 

*  Del  moto  del  gravi  (Florence,  1644). 


ART.  22 


VELOCITY  OF  FLOW  FROM  ORIFICES 


49 


FIG.  22 


The  theoretic  velocity  of  flow  from  the  orifice  is  the 
same  as  that  acquired  by  a  body  falling  from  rest  in  a 
vacuum  through  a  height  equal  to  the  head  of  water  on 
the  orifice. 

One  proof  of  this  theorem  is  by  experience.  If  a  vessel  be 
arranged,  as  in  the  first  diagram  of  Fig.  22,  so  that  a  jet 
of  water  from  an  orifice  is 
directed  vertically  upward, 
it  is  known  that  it  never 
attains  to  the  height  of  the 
level  of  the  water  in  the  ves- 
sel, although  under  favor- 
able conditions  it  nearly 
reaches  that  level.  It  may 
hence  be  inferred  that  the 
jet  would  actually  rise  to  that  height  were  it  not  for  the 
resistance  of  the  air  and  the  friction  of  the  edges  of  the 
orifice.  Now,  since  the  velocity  required  to  raise  a  body 
vertically  to  a  certain  Height  is  the  same  as  that  acquired 
by  it  in  falling  from  rest  through  that  height,  it  is  regarded 
as  established  that  the  velocity  at  the  orifice  is  that  stated 
in  the  theorem. 

The  following  proof  rests  on  the  law  of  conservation  of 
energy.  Let,  as  in  the  second  diagram  of  Fig.  22,  the  water 
surface  in  a  vessel  be  at  A  and  let  the  flow  through  the 
orifice  occur  for  a  very  short  interval  of  time  during  which 
the  water  surface  descends  to  A^  Let  W  be  the  weight 
of  water  between  the  planes  A  and  Alt  which  is  evidently 
the  same  as  that  which  flows  from  the  orifice  during  the 
short  time  considered.  Let  W±  be  the  weight  of  water 
between  the  planes  A^  and  B,  and  /^  the  height  of  its  center 
of  gravity  above  the  orifice.  Let  h  be  the  height  of  A 
above  the  orifice,  and  dh  the  small  distance  between  A 
and  Alt  At  the  beginning  of  the  flow  *the  water  in  the 
-vessel  has  the  potential  energy  W^  +  WQi  —  ^dh)  with 


50  THEORETICAL  HYDRAULICS  CHAP,  in 

respect  to  B.  If  V  be  the  velocity  at  the  orifice  the  same 
water  at  the  end  of  the  short  interval  of  time  has  the  energy 
Wfa  +  W  .V2/2g.  By  the  law  of  conservation  these  are 
equal  if  no  energy  has  been  expended  in  overcoming  fric- 
tional  resistances;  thus  h  —  %dh  =  V2/2g.  Here  oh  is  very 
small  if  the  area  A  is  large  compared  with  the  area  of  the 
orifice,  and  thus  V2  =  2gh,  which  is  the  same  as  for  a  body 
falling  from  rest  through  the  height  h.  Or  h  —  ^dh  may 
be  regarded  as  an  average  head  corresponding  to  an  aver- 
age velocity  V,  so  that  in  general  V2/2g  is  equal  to  the 
average  head  on  the  orifice. 

For  any  orifice,  therefore,  whether  its  plane  be  hori- 
zontal, vertical,  or  inclined,  provided  the  head  h  be  so 
large  that  it  has  practically  the  same  value  for  all  parts 
of  the  orifice, 


V  =     2gh  h  =  V2/2g  (22)  L 

the  first  of  which  gives  the  theoretic  velocity  of  now  due 
to  a  given  head,  while  the  second  gives  the  theoretic  head 
that  will  produce  a  given  velocity.  The  term  "  velocity- 
head"  will  generally  be  used  to  designate  the  expression 
V2/2g,  this  being  the  height  to  which  the  jet  would  rise 
if  it  were  directed  vertically  upward  and  there  were  no 
frictional  resistances.  Using  for  g  the  mean  value  32.16 
feet  per  second  per  second  (Art.  7),  these  formulas  become 

V  =  S.o2oVh         h  =  0.01  555  V2  (22)* 

in  which  h  must  be  in  feet  and  V  in  feet  per  second.  Table 
15  gives  values  of  the  velocity  V  corresponding  to  a  given 
head  h  and  also  values  of  the  velocity-head  h  correspond- 
ing to  a  given  velocity  V.  It  is  seen  that  small  heads 
produce  high  theoretic  velocities.  The  relation  between 
h  and  V  is  the  same  as  that  between  the  ordinate  and 
abscissa  of  the  common  parabola  when  the  origin  is  at 
the  vertex.  It  may  also  be  noted  that  the  discussion  here 
given  applies  not  only  to  water  but  to  any  liquid;  thus. 


ART.  23  DISCHARGE    FROM   SMALL   ORIFICES  51 

V2  =  2gh  is  theoretically  true  for  alcohol  and  mercury  as 
well  as  for  water. 

Prob.  22a.  Find  from  Table  15  the  velocity  due  to  a  head 
of  0.05  feet,  and  the  velocity-head  corresponding  to  a  velocity 
of  0.6  feet  per  second. 

Prob.  226.  Compute  the  theoretic  velocity  due  to  a  head  of 
0.064  feet,  and  the  velocity-head  corresponding  to  velocity  of 
0.18  feet  per  second. 

ART.  23.     DISCHARGE  FROM  SMALL  ORIFICES 

The  term  " discharge"  means  the  quantity  of  water 
flowing  in  one  second  from  a  pipe  or  orifice,  and  the  letter 
Q  will  designate  the  theoretic  discharge,  that  is,  the  dis- 
charge as  computed  without  considering  the  losses  due 
to  frictional  resistances.  When  all  the  filaments  of  water 
issue  from  the  pipe  or  orifice  with  the  same  ^velocity,  the 
quantity  of  water  issuing  in  one  second  is  equal  to  the 
volume  of  a  prism  having  a  base  equal  to  the  cross-section 
of  the  stream  and  a  length  equal  to  the  velocity.  If  this 
area  be  a  and  the  theoretic  velocity  be  V,  then  Q=^aV  is 
the  theoretic  discharge.  Taking  a  in  square  feet  and  V 
in  feet  per  second,  Q  is  cubic  feet  per  second. 

For  a  small  orifice  on  which  the  head  h  has  the  same 
value  for  all  parts  of  the  opening,  the  theoretic  discharge  is 

Q=aV  =  aV^gh  (23) 

and  in  English  measures  Q  =  S.o2aVh.  For  example,  let 
a  circular  orifice  3  inches  in  diameter  be  under  a  head 
of  10.5  feet,  and  let  it  be  required  to  compute  Q.  Here 
3  inches  =0.25  feet  and  from  Table  51  the  area  of  the 
circle  is  0.04909  square  feet.  From  Art.  22  the  theoretic 
velocity  Vis  8.02X10.5^  =  25.99  feet  per  second.  Accord- 
ingly the  theoretic  discharge  is  0.04909X25. 99  =-1.28  cubic 
feet  per  second. 

The  above  formula  for  Q  applies  strictly  only  to  hori- 
zontal orifices  upon  which  the  head  h  is  constant,  but  it 


52  THEORETICAL  HYDRAULICS  CHAP,  nr 

will  be  seen  later  that  its  error  for  vertical  orifices  is  less 
than  one  half  of  one  percent  when  h  is  greater  than  double 
the  depth  of  the  orifice.  Horizontal  orifices  are  but  little 
used,  as  it  is  more  convenient  in  practice  to  arrange  an 
opening  in  the  side  of  a  vessel  than  in  its  base.  In  apply- 
ing the  above  formula  to  a  vertical  orifice,  h  is  taken  as 
the  vertical  distance  from  its  center  to  the  free-water  sur- 
face. Vertical  orifices  where  h  is  small  are  discussed  in 
Arts.  47  and  48. 

Since  the  theoretic  velocity  is  always  greater  than  the 
actual  velocity,  the  theoretic  discharge  is  a  limit  which  can 
never  be  reached  under  actual  conditions.  Theoretically 
the  discharge  is  independent  of  the  shape  of  the  orifice,  so 
that  a  square  orifice  of  area  a  gives  the  same  theoretic  dis- 
charge as  a  circular  orifice  of  area  a;  it  will  be  seen  in 
Chapter  V  that  this  is  not  quite  true  for  the  actual  dis- 
charge. 

In  this  chapter  it  is  supposed  that  the  velocity  of  a  jet 
is  the  same  in  all  parts  of  the  cross-section,  as  this  would 
be  the  case  if  h  has  the  same  value  throughout  the  section 
were  it  not  for  the  retarding  influence  of  friction.  Actually, 
however,  the  filaments  of  water  near  the  edges  of  the 
orifice  move  slower  than  those  near  the  center.  If  q  be 
the  actual  discharge  from  any  orifice  and  v  the  mean 
velocity  in  the  area  a,  then  q  =  av,  or  the  equation  v  =  q/a 
may  be  regarded  as  a  definition  of  the  term  ' '  mean  veloc- 
ity. ' '  The  theoretic  mean  velocity  is  S/2g&,  but  the  actual 
mean  velocity  is  slightly  smaller,  as  will  be  seen  in  Chap.  V. 

Formula  (23)  may  be  used  for  computing  h  when  Q 
and  a  are  given,  and  it  shows  that  the  theoretic  head 
required  to  deliver  a  given  discharge  varies  inversely  as. 
the  square  of  the  area  of  the  orifice. 

Prob.  23.  Compute  the  theoretic  head  required  to  deliver 
288  gallons  of  water  per  minute  through  an  orifice  3  inches  in 
diameter.  Compute  also  the  theoretic  velocity. 


ART.  24  FLOW    UNDER    PRESSURE  53 

ART.  24.     FLOW  UNDER  PRESSURE 

The  level  of  water  in  the  reservoir  and  the  orifice  of 
outflow  have  been  thus  far  regarded  as  subjected  to  no 
pressure,  or  at  least  only  to  the  pressure  of  the  atmos- 
phere which  acts  upon  both  with  the  same  mean  force  of 
14.7  pounds  per  square  inch,  since  the  head  h  is  rarely  or 
never  so  great  that  a  sensible  variation  in  atmospheric 
pressure  can  be  detected  between  the  orifice  and  the  water 
level.  But  the  upper  level  of  the  water  may  be  subject 
to  the  pressure  of  steam  or  to  the  pressure  due  to  a  heavy 
weight  or  to  a  piston.  The  orifice  may  also  be  under  a 
pressure  greater  or  less  than  that  of  the  atmosphere.  It 
is  required  to  determine  the  velocity  of  flow  from  the 
orifice  under  these  conditions. 

First,  suppose  that  the  surface  of  the  water  in  the  ves- 
sel or  reservoir  is  subjected  to  the  uniform  pressure  of  pQ 
pounds  per  square  unit  above  the  atmospheric  pressure, 
while  the  pressure  at  the  orifice  is  the  same  as  that  of  the 
atmosphere.  Let  h  be  the  depth  of  water  on  the  orifice. 
The  velocity  of  flow  V  is  greater  than  \/2gh  on  account 
of  the  pressure  p0,  and  it  is  evidently  the  same  as  that  from 
a  column  of  water  whose  height  is  such  as  to  produce  the 
same  pressure  at  the  orifice.  If  w  be  the  weight  of  a  cubic 
unit  of  water  the  unit  -pressure  at  the  orifice  due  to  the 
head  is  wh,  and  the  total  unit-pressure  at  the  depth  of  the 
orifice  is  p=wh-\-p^  and  from  formula  (II)*  the  head  of 
water  which  would  produce  this  total  unit-pressure  is 


w  w 

Accordingly  the  theoretic  velocity  of  flow  from  the  orifice  is 


or,  if  h0  denote  the  head  corresponding  to  the  pressure 


THEORETICAL  HYDRAULICS 


CHAP.  Ill 


The  general  formula  (22)  t  thus  applies  to  any  small  orifice 
if  H  be  the  head  corresponding  to  the  static  pressure  at 
the  orifice. 

Secondly,  suppose  that  the  surface  of  the  water  in  the 
vessel  is  subjected  to  the  unit-pressure  pQ,  while  the  orifice 
is  under  the  external  unit-pressure  p^.  Let  h  be  the  head 
of  actual  ,water  on  the  orifice,  h0  the  head  of  water  which 
will  produce  the  pressure  p0t  and  h1  the  head  which  will 
produce  pv  The  velocity  of  flow  at  the  orifice  is  then  the 
same  as  if  the  orifice  were  under  a  head  h  +  h  —  h  or 


hfco-fci)  (24)i 

in  which  the  values  of  hQ  and  ht  are 

ko=Po/w        h1=pl/w 

Usually  />0  and  p1  are  given  in  pounds  per  square    inch, 
while  h0  and  ht  are  required  in  feet;  then  (Art.  11) 
7*0  =  2.304^0,         ^1  =  2.304^ 

The  values  of  p0  and  p±  may  be  absolute  pressures,  or  merely 
pressures  above  the  atmosphere.  In  the  latter  case  pl  may 
sometimes  be  negative,  as  in  the  discharge  of  water  into  a 
condenser. 

As  an  illustration  of  these  principles  let  the  cylindrical 
tank  in  Fig.  24  be  2  feet  in  diameter,  and  upon  the  surface 

of  the  water  let  there  be  a 
tightly    fitting    piston    which 
with  the  load  W  weighs  3000 
pounds.     At  the  depth  8  feet 
below  the  water  level  are  three 
small  orifices:  one  at  A,  upon 
which  there  is  an  exterior  head 
of  water  of    3   feet;  one  not 
shown  in  the  figure,  which  dis- 
charges directly  into  the  atmosphere ;  and  one  at  C,  where 
the  discharge  is  into  a  vessel  in  which  the  air  pressure  is 
only  10  pounds  per  square  inch.     It  is  required  to  deter- 


ART.  24  FLOW    UNDER    PRESSURE  55 

mine  the  velocity  of  efflux  from  each  orifice.     The  head  hQ 
corresponding  to  the  pressure  on  the  upper  water  surface  is 


p  =  __  2 

w      3.142X62.5 

The  head  h^  is  3  feet  for  the  first  orifice,  o  for  the  second, 
and  -2.304(14.7  —  10)  =  —  10.83  feet  for  the  third.  The 
three  theoretic  velocities  of  outflow  then  are: 


F  =  8.o2\/8  +  15.28  —  3  =36.1  feet  per  second, 
V  =8.o2V/8  +  i5.28—  o  —38.7  feet  per  second, 
F  =  8.  02\/8  +  15.28+  10.83  =46.8  feet  per  second. 

In  the  case  of  discharge  from  an  orifice  under  water,  as 
at  A  in  Fig.  24,  the  value  of  h  —  h^  is  the  same  wherever 
the  orifice  be  placed  below  the  lower  level,  and  hence  the 
velocity  depends  upon  the  difference  of  level  of  the  two 
water  surfaces,  and  not  upon  the  depth  of  the  orifice. 

The  velocity  of  flow  of  oil  or  mercury  under  pressure  is 
to  be  determined  in  the  same  manner  as  water  by  finding 
the  heads  which  will  produce  the  given  pressure.  Thus  in 
the  preceding  numerical  example,  if  the  liquid  be  mercury, 
whose  weight  per  cubic  foot  is  850  pounds,  the  head  of  mer- 
cury corresponding  to  the  pressure  of  the  piston  is 

7  3°°° 


and,  accordingly,  for  discharge  into  the  atmosphere  at  the 
depth  h  =  &  feet  the  velocity  is 

F  =  8.o2\/8  +  1.  1  2  =24.2  feet  per  second, 
while  for  water  the  velocity  was  38.7  feet  per  second.  The 
general  formula  (22)  1  is  applicable  to  all  cases  of  the  flow 
of  liquids  from  a  small  orifice  if  for  h  its  value  p/w  be  sub- 
stituted, where  p  is  the  resultant  unit  -pressure  at  the  depth 
of  the  orifice  and  w  the  weight  of  a  cubic  unit  of  the  liquid. 
Thus  for  any  liquid 

V  =  V2gp/w  (24), 


56  THEORETICAL  HYDRAULICS  CHAP,  in 

is  the  theoretic  velocity  of  flow  from  the  orifice.  Accord- 
ingly for  the  same  unit-pressure  p  the  velocities  are  inversely 
proportional  to  the  square  roots  of  the  densities  of  the  liquids. 

Prob.  24a.  What  is  theoretic  velocity  of  flow  from  a  small 
orifice  in  a  boiler  i  foot  below  the  water  level  when  the  steam- 
gage  reads  60  pounds -per  square  inch?  What  is  the  theoretic 
velocity  when  the  gage  reads  o? 

Prob.  246.  A  vessel  i  foot  square  has  a  small  orifice  in  the 
base.  What  is  the  theoretic  velocity  of  flow  from  the  orifice 
when  the  vessel  contains  125  pounds  of  mercury?  What  is 
the  theoretic  velocity  when  the  vessel  contains  125  pounds  of 
oil  whose  specific  gravity  is  0.75? 


ART.  25.     INFLUENCE  OF  VELOCITY  OF  APPROACH 

Thus  far  in  the  determination  of  the  theoretic  velocity 
and  discharge  from  an  orifice,  the  head  upon  it  has  been 
regarded  as  constant.  But  if  the  cross-section  of  the  vessel 
'is  not  large,  the  head  can  only  be  kept  constant  by  an  inflow 
of  water  and  this  will  modify  the  previous  formulas.  In 
this  case  the  water  approaches  the  orifice  with  an  initial 
velocity.  Let  a  be  the  area  of  the  orifice  and  A  the  area  of 
A  the  horizontal  cross-section  of  the  vessel. 

Let  V  be  the  velocity  of  flow  through  a 
and  v  be  the  vertical  velocity  of  inflow 
through  A.  Let  W  be  the  weight  of 
water  flowing  from  the  orifice  in  one 
second ;  then  an  equal  weight  must  enter 


FIG.  25a  .  .  ,  .  . 

at  A  in  one  second  in  order  to  maintain  a 

constant  head  h.  The  kinetic  energy  of  the  outflowing 
water  is  W.V2/2g  and  this  is  equal,  if  there  be  no  loss  of 
energy,  to  the  potential  energy  Wh  of  the  inflowing  water 
plus  its  kinetic  energy  W  .v2/2g,  or 

V2 


ART.  25  INFLUENCE    OF    VELOCITY    OF    APPROACH  57 

Now  since  the  same  quantity  of  water  Q  passes  through  the 
two  areas  in  one  second,  Q=aV=Av,  whence  v  =  V  .a/A. 
Inserting  this  value  of  v  in  the  equation  of  energy,  there 
is  found 


_ 

which  is  always  greater  than  the  value  V2gh. 

The  influence  of  the  velocity  of  approach  on  the  velocity 
of  flow  at  the  orifice  can  now  be  ascertained  by  assigning 
values  to  the  ratio  a/  A.  Thus,  if  a=A  the  velocity  V 
must  be  infinite  in  order  that  the  water  may  fill  the  entire 
section  of  the  vessel  and  orifice.  Further, 

for  a=   |A  V 

for  a=   %A  V 

for  a=   %A  V 

for  a=  iA  V 

for  a 


It  is  here  seen  that  the  common  formula  (22^  is  in  error  2  3 
percent  when  a=fA,  if  the  head  be  maintained  constant 
by  a  uniform  vertical  inflow  at  the  water  surface,  and  0.5 
percent  when  a  =  -fa  A  .  Practically,  if  the  area  of  the  ori- 
fice be  less  than  one  twentieth  of  the  cross-section  of  the 
vessel,  the  error  in  using  the  formula  V  =  \/2gh  is  too 
small  to  be  noticed  even  in  the  most  precise  experiments, 
and  fortunately  most  orifices  are  smaller  in  relative  size 
than  this. 

A  more  common  case  is  that  where  the  reservoir  is  of 
large  horizontal  and  small  vertical  cross-section,  and  where 
the  water  approaches  the  orifice  with  velocity  in  a  hori- 
zontal direction,  as  in  Fig.  256.  Here  let  A  be  the  area 
of  the  vertical  cross-section  of  the  trough  or  pipe,  a  the 
area  of  the  orifice  and  h  the  head  on  its  center.  Then  if 
h  be  large  compared  with  the  depth  of  the  orifice,  exactly 
the  same  reasoning  applies  as  before  and  the  theoretic 


58  THEORETICAL  HYDRAULICS  CHAP,  in 

velocity  at  the  orifice  is  given  by  the  above  formula  (25) t. 
The  same  is  also  true  for  the  case  shown  in  Fig.  25c,  where 


FIG.  256  FIG.  25c 

water  is  forced  through  a  hose  with  the  velocity  v  and 
issues  from  a  nozzle  with  the  velocity  V,  the  head  h  being 
that  due  to  the  pressure  at  the  entrance  of  the  nozzle. 

The  '  '  effective  head  "  on  an  orifice  is  the  head  that  will 
produce  the  theoretic  velocity  V.  If  H  be  this  effective 
head,  then  H  =  V2/2g,  and  from  the  first  equation  of  this 
article  its  value  may  be  written 


(25) 


The  effective  head  on  an  orifice  is,  therefore,  the  sum  of 
the  pressure  and  velocity  heads  which  exist  behind  it. 
Another  expression  for  the  effective  head  can  be  obtained 
from  formula  (25) v  namely, 

H=i-(a/A)* 

When  H  has  been  found  from  either  of  these  formulas  the 
theoretic  velocity  and  discharge  are  given  by 

and         0=aV  = 


for  all  instances  where  h  is  sufficiently  large  so  that  its  value 
is  sensibly  constant  for  all  parts  of  the  orifice.  But  if  this 
is  not  the  case  the  value  of  Q  is  to  be  found  by  the  methods 

of  Arts.  47  and  48. 

Prob.  25.  In  Fig.  25c  let  the  head  h  be  50  feet,  the  diameter 
of  the  nozzle  ij  inches,  and  the  diameter  of  the  hose  3  inches. 
Compute  the  effective  head  H,  and  also  the  discharge  Q  in 
cubic  feet  per  second. 


ART.  26  EMPTYING    A   VESSEL  59 


ART.  26.     EMPTYING  A  VESSEL 

Let  the  depth  of  water  in  a  vessel  be  H ;  it  is  required 
to  determine  the  theoretic  time  of  emptying  it  through  an 
orifice  in  the  base  whose  area  is  a.  Let  Y  be  ,  , 

the  area  of  the  water  surface  when  the  depth 
of  water  is  y\   let  dt  be  the  time  during  which 
the  water  level  falls  the  distance  'dy.     During 
this  time  the  quantity  of  water  Y  .dy  passes 
through  the  orifice.     But  the  discharge  in  one      FIG.  26a 
second  under  the  constant  head  y  is  a\/2gyy  and  hence  the 
discharge  in  the  time  dt  is  adtV2gy.     Equating  these  two 
expressions,   there   is    found    the    general   formula   which 
gives  the  time  for  the  water  surface  to  drop  the  distance  dy. 


(26) 


The  time  of  emptying  any  vessel  is  now  determined  by 
inserting  for  Y  its  value  in  terms  of  y,  and  then  integrating 
between  the  limits  H  and  o. 

For  a  cylinder  or  prism  the  cross-section   Y  has  the 
constant  value  A,  and  the  formula  becomes 


aV2g 
the  integration  of  which,  between  limits  H  and  h,  gives 


av  2g 

as  the  theoretic  time  for  the  head  H  to  fall  to  h.  If  h  =or 
this  formula  gives  the  time  of  emptying  the  vessel.  If 
the  head  were  maintained  constant  the  uniform  discharge 
per  second  would  be  a\/2gH,  and  the  time  of  discharging 
a  quantity  equal  to  the  capacity  of  the  vessel  is  AH  divided 
by  aV2gH,  which  is  one  half  of  the  time  required  to  empty  it. 


60  THEORETICAL  HYDRAULICS  CHAP,  in 

To  find  the  time  of  emptying  a  hemispherical  bowl  of 
radius  r  through  a  small  orifice  at  its  lowest  point,  let  x 
be  the  radius  of  the  cross-section  Y\  then  x2+(r  —  y)2=r2 
is  the  equation  of  the  circle,  from  which  the  area  Y  is 
—  y2).  The  general  formula  then  becomes 


aV  2g 
and  by  integration  between  the  limits  r  and  o 


which  is  the  theoretic  time  required  to  empty  the  hemi- 
spherical bowl. 

The  only  important  application  of  these  principles  is 
in  the  case  of  the  right  prism  or  cylinder,  and  here  the 
formula  for  the  time  is  modified  in  practice  by  introduc- 
ing a  coefficient,  as  may  be  seen  in  Art.  58.  The  theoretic 
time  found  by  the  above  formula  is  always  too  small, 
since  frictional  resistances  have  not  been  considered. 
Moreover,  the  formula  does  not  strictly  apply  when  the 
head  is  very  small  owing  to  a  whirling  motion  that  occurs 
and  which  tends  to  increase  the  theoretic  time. 

Venturi,  in  1798,  first  described  the  phenomena  of  this 
whirl.*  When  the  head  becomes  less  than  about  three 
diameters  of  the  orifice  the  water  is  observed  in  whirling 
motion,  the  velocity  being  greatest  near  the  vertical  axis 
through  the  center  of  the  orifice,  and  as  the  head  decreases 
a  funnel  is  formed  through  the  middle  of  the  issuing  stream. 
The  direction  of  this  whirl,  as  seen  from  above,  may  be 
either  clockwise  or  contraclockwise,  depending  on  initial 
motions  in  the  water  or  on  irregularities  in  the  vessel  or 
orifice,  but  under  ideal  conditions  it  should  be  clockwise 
in  the  southern  hemisphere  of  the  earth  and  contraclock- 

*Tredgold's   Tracts   on    Hydraulics  (London,   1799  and   1826)  gives  a 
translation  of  trie  memoir  of  Venturi. 


ART.  26 


EMPTYING  A  VESSEL 


61 


FIG.  266 


wise  in  the  northern  hemisphere,  this  being  the  effect  of 
the  earth's  rotation.  Fig.  266  represents  a  vertical  sec- 
tion of  this  funnel,  on  which  A 
is  any  point  having  the  coordi- 
nates x  and  y  with  respect  to  the 
rectangular  axes  OX  and  OY. 
The  axis  OF  is  drawn  through 
the  center  of  the  orifice,  and  OX 
is  tangent  to  the  level  water  sur- 
face at  a  distance  H  above  the 
bottom  of  the  vessel.  Let  r  be  the  radius  of  the  funnel  in 
the  plane  of  the  orifice.  It  is  required  to  find  the  relation 
between  ,r,  y,  H,  and  r,  or  the  equation  of  the  curve  shown 
in  the  figure. 

An  approximate  solution  may  be  made  by  supposing 
that  the  particle  of  water  at  A  is  moving  nearly  horizon- 
tally around  the  axis  0  Y  with  the  velocity  v ;  this  velocity 
must  be  due  to  the  head  y,  whence  v2  =  zgy.  This  particle 
is  acted  upon  by  the  downward  force  AB,  due  to  gravity, 
and  by  the  horizontal  force  AC,  due  to  centrifugal  action, 
and  they  are  proportional  to  g  and  v2/x,  these  being  the 
accelerations  due  to  gravity  and  centrifugal  force.  The 
ratio  AC/AB  is  the  tangent  of  the  angle  0  which  the  water 
surface  at  A  makes  with  the  axis  OX,  for  this  surface  must 
be  normal  to  the  resultant  AD  of  the  two  forces  AB  and 
AC.  When  the  ordinate  y  is  increased  to  y  +  dy  the  ab- 
scissa x  is  decreased  to  x  -  dx,  and  hence  the  value  of  tan# 
must  be  the  same  as  —dy/dx.  Accordingly 

AC      v2       y         dy 
tan0=-r^=— =2^-  =  - 

AB     gx       x          dx 

and  the  integration  of  this  differential  equation  gives 
y  =  C/x2,  in  which  C  is  the  constant  of  integration.  When 
y  equals  H,  the  value  of  x  is  r,  and  hence  C  =  Hr2,  and  thus 

y=Hr2/x2 


62  THEORETICAL  HYDRAULICS  CHAP,  in 

is  the  equation  of  the  curve,  which  may  be  called  a  quad- 
ratic hyperbola,  the  surface  of  the  funnel  being  then  a 
quadratic  hyperboloid.  This  equation  represents  the  curve 
at  one  instant  only,  for  H  continually  decreases  as  the 
water  flows  out,  since  the  direction  of  v  is  not  quite  hori- 
zontal as  the  investigation  assumes.  The  general  phe- 
nomena are,  however,  well  explained  by  this  discussion. 

Prob.  26a.  A  prismatic  vessel  has  a  cross-section  of  18 
square  feet,  and  an  orifice  in  its  base  has  an  area  of  0.18  square 
feet.  Show  that  the  theoretic  time  for  the  water  level  to  drop 
7  feet,  when  the  head  upon  the  orifice  at  the  beginnning  is  16 
feet,  is  24.9  seconds. 

Prob.  266.  Why  does  the  rotation  of  the  earth  tend  to  cause 
a  counterclockwise  rotation  of  the  water  when  a  vessel  is  emptied 
in  the  northern  hemisphere  ?  Make  experiments  on  this  phenom- 
enon in  a  stationary  wash-basin,  and  explain  why  the  funnel 
does  not  form  when  a  large  piece  of  cardboard  floats  on  the 
water  surface  above  the  orifice. 


ART.  27.     THE  PATH  OF  A  JET 

When  a  jet  of  water  issues  from  a  small  orifice  in  the 
vertical  side  of  a  vessel  or  reservoir,  its  direction  at  first 
is  horizontal,  but  the  force  of  gravity 
immediately  causes  the  jet  to  move 
^  in  a  curve  which  will  be  shown  to  be 

jg    / .rir_  l_  the  common  parabola.     Let  x  be  the 

abscissa  and  y  the  ordinate  of  any 
point  of  the  curve,  measured  from 
the  orifice  as  an  origin,  as  seen  in 
Fig.  27a.  The  effect  of  the  impulse 
at  the  orifice  is  to  cause  the  space  x 

to  be  described  uniformly  in  a  certain  time  t,  or,  if  v  be 
the  velocity  of  flow,  x  =  vt.  The  effect  of  the  force  of 
gravity  is  to  cause  the  space  y  to  be  described  in  accord- 
ance with  the  laws  of  falling  bodies  (Art.  21),  or  y  = 


ART.  27  THE    PATH    OF    A    JET  63 

Eliminating  t  from  these  two  equations  and  replacing  v2 
by  its  theoretic  value  2gh,  gives 


which  is  the  equation  of  a  parabola  whose  axis  is  vertical 
and  whose  vertex  is  at  the  orifice. 

The  horizontal  range  of  the  jet  for  any  given  ordinate  y 
is  found  from  the  equation  x2  =  ^hy.  If  the  height  of  the 
vessel  be  /,  the  horizontal  range  on  the  plane  of  the  base  is 


This  value  is  o  when  h  =  o  and  also  when  h=l,  and  it  is  a 
maximum  when  h  =  \l.  Hence  the  greatest  range  is  from 
an  orifice  at  the  mid-height  of  the  vessel. 

A  more  general  case  is  that  where  the  side  of  the  vessel 
is  inclined  to  the  vertical  at  the  angle  6,  as  in  Fig.  27b. 
Here  the  jet  at  first  issues 
perpendicularly  to  the  side 
with  a  velocity  v  having  the 
theoretic  value  V2gh,  and 
under  the  action  of  the  impul- 
sive force  a  particle  of  water 
would  describe  the  distance  * 
AB  in  a  certain  time  t  with  FIG.  276 

the  uniform  velocity  v.  But  in  that  same  time  the  force 
of  gravity  causes  it  to  descend  through  the  distance  EC. 
Now  let  x  be  the  horizontal  abscissa  and  y  the  vertical 
ordinate  of  the  point  C  measured  from  the  origin  A.  Then 
AB=xsecd,  and  BC=xtand  —  y.  Hence 

xsecd=vt,         xtB,n6  —  y  =  %gt2 

The  elimination  of  t  from  these  expressions  gives,  after 
replacing  v2  by  its  value  2gh, 

which  is  also  the  equation  of  a  common  parabola. 


64  THEORETICAL  HYDRAULICS  CHAP,  in 

To  find  the  horizontal  range  in  the  level  of  the  orifice 
make  y  =  o  in  the  last  equation  ;   then 


This  is  o  when  6  =  o°  or  6  =  90°  ;  it  is  a  maximum  and  equal 
to  2h  when  #=45°.  To  find  the  highest  point  of  the  jet 
the  first  derivative  of  y  with  reference  to  x  is  to  be  equated 
to  zero  in  order  to  obtain  the  maximum  ordinate,  and  by 
this  method  there  results 


which  are  the  coordinates  of  the  highest  point  with  respect 
to  the  origin  A.  In  these  if  6  =  90°,  x  is  o  and  y  is  h,  that 
is,  if  a  jet  be  directed  vertically  upward  it  will,  theoretically, 
rise  to  the  height  of  the  water  level  in  the  reservoir. 

As  a  numerical  example  let  a  vessel  whose  height  is  16 
feet  stand  upon  a  horizontal  plane  DE,  Fig.  276,  the  side  of 
the  vessel  being  inclined  to  the  vertical  at  the  angle  6  =  30°. 
Let  a  jet  issue  from  a  small  orifice  at  A  under  a  head  of  10 
feet.  The  jet  rises  to  its  maximum  height,  y  =  JXio  =  2.5 
feet,  at  the  distance  x  =  J\/3  X  10  =  8.66  feet  from  A.  At 
#  =  17.32  feet  the  jet  crosses  the  horizontal  plane  through 
the  orifice.  To  locate  the  point  where  it  strikes  the  plane 
DE,  the  value  of  y  is  made  —  6  feet  ;  then,  from  the  equa- 
tion of  the  curve,  x  is  found  to  be  24.6  feet,  whence  the 
distance  DE  is  21.2  feet. 

In  practice  the  above  equations  are  modified  by  the 
frictional  resistance  of  the  edges  of  the  orifice  which  ren- 
ders v  less  than  the  theoretic  value  \/2gh,  and  also  by  the 
resistance  of  the  air.  They  are,  indeed,  extreme  limits 
which  may  be  approached  but  not  reached  by  equations 
that  take  these  resistances  into  account. 

Prob.  27.  A  jet  issues  from  a  vessel  under  a  head  of  6  feet, 
the  side  of  the  vessel  being  inclined  to  the  vertical  at  an  angle 
of  45°  and  its  depth  being  8  feet.  Find  the  maximum  height 


ART.  28  THE    ENERGY    OF   A   JET  65 

to  which  the  jet  rises  and  the  point  where  it  strikes  the  hori-. 
zontal  plane  of  the  base.  Show  that  its  theoretic  velocity  as 
it  strikes  that  plane  is  22.7  feet  per  second. 


ART.  28.     THE  ENERGY  OF  A  JET 

Let  a  jet  or  stream  of  water  have  the  velocity  v,  and  let 
W  be  the  weight  of  water  per  second  passing  any  given 
cross-section.  The  kinetic  energy  of  this  moving  water 
is  the  same  as  that  stored  up  by  a  body  falling  freely  tinder 
the  action  of  gravity  through  a  height  h  and  thereby  ac- 
quiring the  velocity  v.  Thus,  if  K  be  its  kinetic  energy 


Therefore,  for  a  constant  quantity  of  water  per  second 
passing  through  the  given  cross-section,  the  energy  of  the 
jet  is  proportional  to  the  square  of  its  velocity.  If  this 
energy  can  all  be  transformed  into  useful  work,  the  work 
that  the  jet  will  perform  in  one  second  is  Wh. 

The  weight  W,  however,  may  be  expressed  in  terms  of 
the  cross-section  of  the  jet  and  its  velocity.  Thus,  if  a 
be  the  area  of  the  cross-section,  and  w  the  weight  of  a  cubic 
unit  of  water,  W  is  the  weight  of  a  column  of  water  whose 
length  is  v  and  whose  cross-section  is  a,  or  W  =wav\  and 
hence  the  above  formula  reduces  to 

K  =  wav*/2g  (28), 

In  general,  then,  it  may  be  stated  that  for  a  constant  cross- 
section,  the  energy  of  a  jet,  or  the  work  which  it  is  capable 
of  doing  per  second,  varies  with  the  cube  of  its  velocity. 

The  expressions  just  deduced  give  the  theoretic  energy 
of  the  jet,  that  is,  the  maximum  work  which  can  be  obtained 
from  it;  but  this  in  practice  can  never  be  fully  utilized. 
The  amount  of  work  which  is  realized  when  a  jet  strikes 
a  moving  surface,  like  the  vane  of  a  water-motor,  depends 


66  THEORETICAL  HYDRAULICS  CHAP,  in 

upon  a  number  of  circumstances  which  will  be  explained 
in  a  later  chapter,  and  it  is  the  constant  aim  of  inventors 
so  to  arrange  the  conditions  that  the  actual  work  may  be 
as  near  to  the  theoretic  energy  as  possible.  The  "  effi- 
ciency" of  an  apparatus  for  utilizing  the  energy  of  moving 
water  is  the  ratio  of  the  work  actually  utilized  to  the 
theoretic  work;  or,  if  k  be  the  work  realized,  the  efficiency 
e  is  e  =  k/K  (28), 

The  greatest  possible  value  of  e  is  unity,  but  this  can  never 
be  attained,  owing  to  the  imperfections  of  the  apparatus 
and  the  frictional  resistances.  Values  greater  than  0.90 
have,  however,  been  obtained;  that  is,  90  percent  or  more 
of  the  theoretic  energy  of  the  water  has  been  utilized  in 
some  of  the  best  forms  of  hydraulic  motors. 

For  example,  let  water  issue  from  a  pipe  2  inches  in 
diameter  with  a  velocity  of  10  feet  per  second.  The 
cross-section  in  square  feet  is  3.142/144,  and  the  kinetic 
energy  of  the  jet  in  foot-pounds  per  second  is 


=  0.01555 

which  is  0.0385  horse-powers.  If  the  velocity  is  100  feet 
per  second,  the  theoretic  horse-power  will  be  38.5;  if 
this  jet  operates  a  moter  yielding  27.7  effective  horse- 
powers, the  efficiency  of  the  apparatus  is  27.7/38.5=0.72, 
or  72  percent  of  the  theoretic  energy  is  utilized. 

The  energy  of  a  jet  is  the  same  whether  its  direction 
of  motion  be  vertical,  horizontal,  or  inclined,  and  its 
energy  per  second  is  always  Wh,  where  h  is  the  velocity- 
head  corresponding  to  actual  velocity  v,  and  W  is  the 
weight  of  water  delivered  per  second.  The  energy  should 
not  be  computed  from  the  theoretical  velocity  V,  as  this 
is  usually  greater  than  the  actual  velocity. 

Prob.  28a.  A  small  turbine  wheel,  using  102  cubic  feet  of 
water  per  minute  under  a  head  of  40  feet,  is  found  to  give  5.5 
horse-powers.  Find  the  efficiency  of  the  wheel. 


ART.  29  IMPULSE    AND    REACTION    OF    A    JET  67 

Prob.  286.  When  water  issues  from  a  pipe  with  a  velocity 
of  6  feet  per  second  its  kinetic  energy  is  sufficient  to  generate 
1.3  horse-powers.  What  is  the  horse-power  when  the  velocity 
becomes  12  feet  per  second? 

ART.  29.     IMPULSE  AND  REACTION  OF  A  JET 

When  a  stream  or  jet  is  in  motion  delivering  W  pounds 
of  water  per  second  with  the  uniform  velocity  v,  that 
motion  may  be  regarded  as  produced  by  a  constant  force 
F,  which  has  acted  upon  W  for  one  second  and  then 
ceased.  In  this  second  the  velocity  of  W  has  increased 
from  o  to  v,  and  the  space  %v  has  been  described.  Con- 
sequently the  work  F  X  \v  has  been  imparted  to  the  water 
by  the  force  F.  But  the  kinetic  energy  of  the  moving 
water  is  W.v*/2g,  and  hence  by  the  law  of  conservation 
of  energy  FX$v  =  WXv*/2g,  from  which  the  constant 
force  is 

F-Wj  (29)! 

This  value  of  F  is  called  the  "impulse"  of  the  jet.  As 
W  is  in  pounds  per  second,  v  in  feet  per  second,  and  g 
in  feet  per  second  per  second,  the  value  of  F  is  in  pounds 

In  theoretical  mechanics  the  term  "impulse"  is  used 
in  a  slightly  different  sense,  namely,  as  force  multiplied 
"by  time.  In  hydraulics,  however,  W  is  not  pounds,  but 
pounds  per  second,  and  thus  the  impulse  is  simply  pounds. 
The  force  F  is  to  be  regarded  as  a  continuous  impulsive 
pressure  acting  in  the  direction  of  the  motion.  For,  by 
the  definition,  F  acts  for  one  second  upon  the  W  pounds 
of  water  which  pass  a  given  section;  but  in  the  next 
second  W  pounds  also  pass  the  section,  and  the  same 
is  the  case  for  each  second  following.  This  impulse  will 
be  exerted  as  a  pressure  upon  any  surface  placed  in  the 
path  of  the  jet. 

The  reaction  of  a  jet  upon  a  vessel  occurs  when  water 
flows  from  an  orifice.  This  reaction  must  be  equal  in 


68  THEORETICAL  HYDRAULICS  CHA*>.  in 

value  and  opposite  in  direction  to  the  impulse,  as  in  all 
cases  of  stress  action  and  reaction  are  equal.  In  the 
direction  of  the  jet  the  impulse  produces  motion,  in  the 
opposite  direction  it  produces  an  equal  pressure  which 
tends  to  move  the  vessel  backward.  The  force  of  reaction 
of  a  jet  is  hence  equal  to  the  impulse  but  opposite  in 
direction.  For  example  (Fig.  29),  let  a  vessel  containing 
water  be  suspended  at  A  so  that  it  can  swing  freely,  and 
let  an  orifice  be  opened  in  its  side  at  B. 
The  head  of  water  at  B  causes  a  pressure 
which  acts  toward  the  left  and  causes  W 
pounds  of  water  to  move  during  every 
second  with  the  velocity  of  v  feet  per 
second,  and  which  also  acts  toward  the 
FIG.  29  right  and  causes  the  vessel  to  swing  out 

of  the  vertical;  the  first  of  these  forces  is  the  impulse 
and  the  second  is  the  reaction  of  the  jet.  If  a  force  R 
be  applied  on  the  right  of  a  vessel  so  as  to  prevent  the 
swinging,  its  value  is 

R=F  =  W.v/g  (29)2 

and  this  is  the  reaction  of  the  jet. 

The  impulse  or  reaction  of  a  jet  issuing  from  an  orifice 
is  double  the  hydrostatic  pressure  on  the  area  of  the 
orifice.  Let  h  be  the  head  of  water,  a  the  area  of  the 
orifice,  and  w  the  weight  of  a  cubic  unit  of  water;  then, 
by  Art.  15,  the  normal  pressure  when  the  orifice  is  closed 
is  wah.  When  the  orifice  is  opened  the  weight  of  water 
issuing  per  second  is  W =wav,  and  hence  the  impulse 
or  reaction  of  the  jet  is 

R=F==  wav .  v/g  =  2wa .  v*/2g  =  2wah 

which  is  double  the  hydrostatic  pressure.  This  theo- 
retic conclusion  has  been  verified  by  many  experiments. 
(Art.  144.) 

When  a  jet  impinges  normally  on  a  plane  it  produces 
a  dynamic  pressure  on  that  plane  equal  to  the  impulse 


ART.  30          ABSOLUTE  AND  RELATIVE  VELOCITIES  69 

F,  since  the  force  required  to  stop  W  pounds  of  water 
in  one  second  is  the  .same  as  that  required  to  put  it  in 
motion.  Again,  if  a  stream  moving  with  the  velocity  v, 
is  retarded  so  that  its  velocity  becomes  v2,  the  impulse 
in  the  first  instant  is  W  .v^/g  and  in  the  second  W  .v2/g. 
The  difference  of  these,  or 

Fi-Ft-Wfa-vJ/g  (29). 

is  a  measure  of  the  dynamic  pressure  which  has  been 
developed.  It  is  by  virtue  of  the  pressure  due  to  change 
of  velocity  that  turbine  wheels  and  other  hydraulic  motors 
transform  the  kinetic  energy  of  moving  water  into  useful 
work. 

Prob.  29.  If  a  stream  of  water  3  inches  in  diameter  issues 
from  an  orifice  in  a  direction  inclined  downward  26°  to  the 
horizon  with  a  velocity  of  15  feet  per  second,  show  that  its 
upward  reaction  on  the  vessel  is  9.4  pounds  and  that  its  hori- 
zontal reaction  on  the  vessel  is  19.3  pounds.  Show  that  the 
pressure  exerted  by  this  stream,  when  stopped  by  a  plane  nor- 
mal to  its  direction,  is  21.5  pounds. 


ART.  30.     ABSOLUTE  AND  RELATIVE  VELOCITIES 

Absolute  velocity  is  defined  in  this  book  as  that  with 
respect  to  the  surface  of  the  earth,  and  relative  velocity 
as  that  with  respect  to  a  body  moving  on  the  earth.  Thus 
absolute  velocity  is  that  seen  by  a  spectator  who  is  on 
the  earth  and  relative  velocity  is  that  seen  by  one  who 
is  on  the  moving  body.  For  instance,  if  a  body  be  dropped 
by  a  person  who  is  on  a  moving  railroad  car  it  appears  to 
a  person  standing  outside  to  move  obliquely,  but  to  one 
on  the  car  it  appears  to  move  vertically.  On  a  car  in 
uniform  motion  all  the  laws  of  mechanics  prevail  exactly 
as  if  it  were  at  rest,  hence  if  dropped  through  a  height 
h  the  body  acquires  a  theoretic  velocity  of  \/2gh  with 
respect  to  the  car.  But  if  the  velocity  of  the  car  be  M 
the  kinetic  energy  of  the  body  at  the  moment  of  letting 


70  THEORETICAL  HYDRAULICS  CHAP,  in 

it  fall  is  W  .u2/2g  and  its  potential  energy  is  Wh,  so  that, 
neglecting  frictional  resistances,  its  total  energy  as  it 
reaches  the  earth  is  the  sum  of  these  and  accordingly 
its  absolute  velocity  as  it  reaches  the  earth  is  \/2gh  +  u'i. 

When  a  vessel  containing  water  with  a  free  surface,  as 
in  Fig.  30a,  has  an  orifice  under  the  head  h  and  is  in  motion 
in  a  straight  line  with  the  uniform  absolute  velocity  u,  the 

theoretic  velocity  of  flow  rela- 
tive to  the  vessel  is  V  =  \/2gh, 
or  the  same  as  its  absolute  ve- 
locity if  the  vessel  were  at  rest, 
for  no  accelerating  forces  exist 
to  change  the  direction  or  the 
value  of  g.  The  absolute  ve- 
locity of  flow,  however,  may  be  greater  or  less  than  V, 
depending  upon  the  value  of  u  and  its  direction.  To  illus- 
trate, take  the  case  of  a  vessel  in  uniform  horizontal  motion 
from  which  water  is  flowing  through  three  orifices.  At  A 
the  direction  of  V  is  horizontal,  and  as  the  vessel  is  moving 
in  the  opposite  direction  with  the  velocity  u,  the  absolute 
velocity  of  the  water  as  it  leaves  the  orifice  is  v  =  V  —  u. 
It  is  also  plain,  if  the  orifice  were  in  front  of  the  vessel  and 
the  direction  of  V  horizontal,  that  the  absolute  velocity 
of  the  water  as  it  leaves  the  orifice  isv  =  V  +  u. 

Again,  at  B  is  an  orifice  from  which  the  water  issues 
vertically  with  respect  to  the  vessel  with  the  relative  velocity 
V,  while  at  the  same  time  the  orifice  moves  horizontally 
with  the  absolute  velocity  u.  Forming  the  parallelogram, 
the  absolute  velocity  v  is  seen  to  be  the  resultant  of  the 
velocities  V  and  u,  or 


Lastly,  at  C  is  shown  an  orifice  in  the  front  of  the  vessel  so 
arranged  that  the  direction  of  the  relative  velocity  V  makes 
an  angle  <j>  with  the  horizontal.  From  C  draw  Cu  to  rep- 
resent the  velocity  u,  and  CV  to  represent  V,  and  complete 


ART.  30          ABSOLUTE  AND  RELATIVE  VELOCITIES  71 

the  parallelogram  as  shown;  then  Cv,  the  resultant  of  u 
and  V,  is  the  absolute  velocity  with  which  the  water  leaves 
the  orifice.  From  the  triangle  Cuv 


(30) 

In  this,  if  <£  =  o,  the  absolute  velocity  v  becomes  V +  u  as 
before  shown  for  an  orifice  in  the  front;  if  ^  =  90°,  it  be- 
comes the  same  as  when  the  water  issues  vertically  from 
the  orifice  in  the  base;  and  if  <£  =  i8o°,  the  value  of  v  is 
V  —  u  as  before  found  for  an  orifice  in  the  rear  end. 

Another  case  is  that  of  a  revolving  vessel  having  an 
opening  from  which  the  water  issues  horizontally  with  the 
relative  velocity  V,  while  the  orifice  is 
moving  horizontally  with  the  absolute 
velocity  u.  Fig.  306  shows  this  case, 
ft  being  the  angle  wrhich  V  makes  with 
the  reverse  direction  of  u,  and  here  also 


v  =  VV2  +  u2  —  2uV  cos/? 

FIG.  306 

is  the  absolute  velocity  of  the  water  as  it  leaves  the  vessel. 
In  all  cases  the  absolute  velocity  of  a  body  leaving  a  mov- 
ing surface  is  the  diagonal  of  a  parallelogram  one  side  of 
which  is  the  velocity  of  the  body  relative  to  the  surface 
and  the  other  side  is  the  absolute  velocity  of  that  surface. 

If  a  vessel  move  with  a  motion  which  is  accelerated  or 
retarded,  this  affects  the  value  of  g,  and  the  reasoning  of 
the  preceding  articles  does  not  give  the  correct  value  of  V. 
For  instance,  if  a  vessel  move  vertically  upward  with  an 
acceleration  /,  the  relative  velocity  of  flow  from  an  orifice, 
in  it  is  V  =  \/2(g  +  f)ti,  and  if  u  be  the  velocity  of  the  vessel 
at  any  instant,  the  absolute  downward  velocity  of  flow  is 
u  +  V.  Again,  if  a  vessel  be  moving  downward  with  the 
acceleration  /,  the  relative  velocity  of  flow  is  V  =  \/ 2(g  —  f)h 
and  the  absolute  is  u  —  V.  If  the  downward  acceleration 
be  gj  or  the  vessel  be  freely  falling,  V  will  be  zero,  since 


72  THEORETICAL  HYDRAULICS  CHAP,  nr 

both  vessel  and  water  are  alike  accelerated  and  there  is 
no  pressure  on  the  base. 

Prob.  30.  In  Fig.  30a  let  the  orifice  at  A  be  under  a  head 
of  4  feet  and  its  height  above  the  earth  be  4.5  feet,  while 
the  car  moves  with  a  velocity  of  40  miles  per  hour.  Compute 
the  relative  velocity  V,  the  absolute  velocity  v,  and  the  absolute 
velocity  of  the  jet  as  it  strikes  the  earth. 

ART.  31.     FLOW  FROM  A  REVOLVING  VESSEL 

Water  in  a  vessel  at  rest  on  the  surface  of  the  earth  is 
acted  upon  only  by  the  vertical  force  of  gravity,  and  hence 
its  surface  is  a  horizontal  plane.  Water  in  a  revolving 
vessel  is  acted  upon  by  centrifugal  force  as  well  as  by 
gravity,  and  it  is  observed  that  its  surface  assumes  a  curved 
shape.  The  simplest  case  is  that  of  a  cylindrical  vessel 
rotating  with  uniform  velocity  about  its  vertical  axis,  and 
it  will  be  shown  that  here  the  water  surface  is  that  of  a 
paraboloid. 

Let  BC  be  the  vertical  axis  of  the  vessel,  h  the  depth, 
of  water  in  it  when  at  rest,  and  h^  and  h2  the  least  and 
D       greatest  depths  of  water  in  it  when 
in  motion.     Let  G  be  any  point  on 
2    the  surface  of  the  water  at  the  hori- 
zontal distance  x  from  the  axis,  and 


B          E  let  y  be  the  vertical  distance  of  G 

FIG.  sia  above  the  lowest  point  C.     The  head 

of  water  on  any  point  E  in  the  base  is  EG  or  hl  +  y.  Now 
this  head  y  is  caused  by  the  velocity  u  with  which  the  point 
G  revolves  around  the  axis,  or,  in  other  words,  the  position 
of  G  above  C  is  due  to  the  energy  of  rotation.  Thus  if  W 
be  the  weight  of  a  particle  of  water  at  G  the  potential  energy 
Wy  equals  the  kinetic  energy  Wuz/2g,  and  hence  y=u2/2g. 

Let  n  be  the  number  of  revolutions  made  by  the  vessel 
and  water  in  one  second.     Then  u  =  2nx.  n,  and  hence 

y  =  u2/2g  =  2X*n2X2 '/g 


ART.  31  FLOW    FROM    A    REVOLVING    VESSEL  73 

which  is  the  equation  of  a  common  parabola  with  respect 
to  rectangular  axes  having  an  origin  at  its  vertex  C.  The 
surface  of  revolution  is  hence  a  paraboloid. 

Since  the  volume  of  a  paraboloid  is  one  half  that  of 
its  circumscribing  cylinder,  and  since  the  same  quantity 
of  water  is  in  the  vessel  when  in  motion  as  when  at  rest,  it 
is  plain  that  in  the  figure  i(/&2  — ^i)  equals  k  —  h^  Conse- 
quently h  —  hl  equals  h2  —  h,  or  the  elevation  of  the  water 
surface  at  D  above  its  original  level  is  equal  to  its  depres- 
sion at  C.  If  r  be  the  radius  of  the  vessel,  the  height  h2  —  hl 
is,  from  the  above  equation,  2x2n2r2/g,  and  hence  the  dis- 
tances h  —  hi  and  h2  —  h  are  each  equal  to  x2n2r2/g.  The 
head  at  the  middle  of  the  base  of  the  vessel  during  the 
motion  is  now  h^  =  h  —  n2n2r2/g  and  the  head  at  any 'point  E 
is  hi+y=h+(2X2-r2)n2n2/g.  . 

The  theoretic  velocity  of  flow  from  the  small  orifice  in 
the  base  is  that  due  to  the  head  h  +  or 


V  =  \/2g(hl 

which  is  less  than  ^/2gh  when  x2  is  less  than  Jr2,  and  greater 
when  x2  is  greater  than  Jr2.  For  example,  let  r  =  i  foot  and 
&  =  3  feet,  then  1^  =  13.9  feet  per  second  when  the  vessel  is 
at  rest.  But  if  it  be  rotating  three  times  per  second  around 
its  axis  with  uniform  speed  the  velocity  from  an  orifice  in 
the  center  of  the  base,  where  x  =  o,  is  3.9  feet  per  second, 
while  the  velocity  from  an  orifice  at  the  circumference  of 
the  base,  where  x  =  i  foot,  is  19.2  feet  per  second.  At  this 
speed  the  water  is  depressed  2.76  feet  below  its  original 
level  at  the  center  and  elevated  the  same  amount  above 
that  level  around  the  sides  of  the  vessel. 

In  the  case  of  a  closed  vessel  where  the  paraboloid  can- 
not form,  the  velocity  of  flow  from  all  orifices,  except  one 
at  the  axis,  is  increased  by  the  rotation.  Thus  in  Fig.  316, 
if  the  vessel  be  at  rest  and  the  head  on  the  base  be  h  the 
velocity  of  flow  from  all  small  orifices  in  the  base  is  \/2gh. 


74  THEORETICAL  HYDRAULICS  CHAP,  in 

But  if  the  vessel  be  revolved  about  the  vertical  axis  BCt 
c         so  that  an  orifice  at  E  has  the  velocity  u  around 
that    axis,    then   the    pressure-head    at    E  is 
,  and  accordingly 


(31) 

is  the  theoretic  velocity  of  flow  from  an  orifice  at  E.  This 
formula  is  an  important  one  in  the  discussion  of  hydraulic 
motors.  Here,  as  before,  the  value  of  u  may  be  expressed 
as  27r.m,  when  x  is  the  distance  of  E  from  the  axis  and 
n  is  the  number  of  revolutions  per  second.  As  an  ex- 
ample, suppose  a  closed  vessel  full  of  water  to  be  revolved 
about  an  axis  120  times  per  minute,  and  it  be  required 
to  find  the  theoretic  velocity  of  flow  from  an  orifice  ij 
feet  from  the  axis,  the  head  on  which  is  4  feet  when  the 
vessel  is  at  rest.  The  velocity  u  is  found  to  be  18.85 
feet  per  second,  and  then  the  theoretic  velocity  of  flow 
from  the  orifice  is  24.8  feet  per  second,  whereas  it  is  only 
16.0  feet  per  second  when  the  vessel  is  at  rest. 

The  velocity  V  in  both  these  cases  is  a  relative 
velocity,  for  the  pressure  at  the  moving  orifice  produces 
a  velocity  with  respect  to  the  vessel.  The  absolute  ve- 
locity, or  that  with  respect  to  the  earth,  is  greater  than 
the  relative  velocity  when  the  stream  issues  from  an 
orifice  in  the  base,  for  the  orifice  moves  horizontally  with 
the  absolute  velocity  u  and  the  stream  moves  downward 
with  the  relative  velocity  V,  and  hence  the  absolute  ve- 
locity of  the  stream  is  W2  +  u2.  When  the  stream  issues 
from  an  orifice  in  the  side  of  the  vessel  upon  which  the 
head  is  h,  formula  (31)  gives  its  relative  velocity  and 
then  the  absolute  velocity  is  found  by  formula  (30). 

Prob.  31a.  For  the  curve  in  Fig.  31a  deduce  the  equation 
y  —  Coc2  by  a  method  of  proof  similar  to  that  used  in  the  latter 
part  of  Art.  26. 

Prob.  316.  A  cylindrical  vessel  2  feet  in  diameter  and  3  feet 
deep  is  three  fourths  full  of  water,  and  is  revolved  about  its 


ART.  32 


STEADY  FLOW  IN  SMOOTH  PIPES 


75 


vertical  axis  so  that  the  water  is  just  on  the  point  of  overflowing 
around  the  upper  edge.  Find  the  number  of  revolutions  per 
minute.  Find  the  relative  velocity  of  flow  from  an  orifice  in 
the  base  at  a  distance  of  0.75  feet  from  the  axis.  Show  that 
the  velocity  from  all  orifices  within  0.707  feet  of  the  axis  is  less 
than  if  the  vessel  were  at  rest. 


ART.  32.     STEADY  FLOW  IN  SMOOTH  PIPES 

When  water  flows  through  a  pipe  of  varying  cross- 
section  and  all  sections  are  filled  with  water,  the  same 
quantity  of  water  passes  each  section  in  one  second.  This 
is  called  the  case  of  steady  flow.  Let  q  be  this  quantity 
of  water  and  let  vlt  v2t  v3  be  the  mean  velocities  in  three 
sections  whose  areas  are  alf  a2,  a3.  Then 

Q—.Qfy  =  a<u  =  ai)  (32} 

This  is  called  the  condition  for  steady  flow,  and  it  shows 
that  the  velocities  at  different  sections  vary  inversely 
as  the  areas  of  those  sections.  If  v  be  the  velocity  at 
the  end  of  the  pipe  where  the  area  is  a,  then  also  q  =  av. 
When  the  discharge  q  and  the  areas  of  the  cross-sections 
have  been  measured,  the  mean  velocities  maybe  computed. 

When  a  pipe  is  filled  with  water  at  rest  the  pressure 
at  any  point  depends  only  upon  the  head  of  water  above 
that  point.  But  when  the  water  is  in  motion  it  is  a  fact 
of  observation  that  the 


pressure  becomes  less  than  | 

that  due  to  the  head.     The  g 

unit-pressure    in    any    case  =- 

may    be    measured    by    the  | 

height  of  a  column  of  water.  E^E^IE="^- 

Thus  if  water  be  at  rest  in 

the  case  shown  in  Fig.  32a, 

and  small  tubes  be  inserted 


FIG.  32a 


at  the  sections  whose  areas  are  at  and  a2,  the  water  will 
rise  in  each  tube  to  the  same  level  as  that  of  the  water 


76  THEORETICAL  HYDRAULICS  CHAP,  in 

surface  in  the  reservoir,  and  the  pressures  in  the  sections 
will  be  those  due  to  the  hydrostatic  heads  Hl  and  H2. 
But  if  the  valve  at  the  right  be  opened  the  water  levels 
in  the  small  tubes  will  sink  and  the  mean  pressures  in 
the  two  sections  will  be  those  due  to  the  pressure  -heads 
7^  and  h2. 

Let  W  be  the  weight  of  water  flowing  in  each  second 
through  each  section  of  the  pipe,  and  let  v^  and  v2  be  the 
mean  velocity  in  the  section  ai  and  a2.  When  the  water 
wras  at  rest  the  potential  energy  of  pressure  in  the  section 
ax  was  VVH1  ;  when  it  is  in  motion  the  energy  in  the  section 
is  the  pressure  energy  Wh1  plus  the  kinetic  energy  W  .i\2/2g. 
If  no  losses  of  energy  due  to  friction  or  impact  have  oc- 
curred, the  energy  in  the  two  cases  must  be  equal.  The 
same  reasoning  applies  to  the  section  a2,  and  hence 


(32), 


These  equations  exhibit  the  law  first  deduced  by  Daniel 
Bernouilli  in  1738,*  and  which  may  be  stated  in  words 
as  follows: 

At  any  section  of  a  tube  or  pipe,  tinder  steady  flow 
without  friction,  the  pressure-head  plus  the  velocity-head 
is  equal  to  the  hydrostatic  head  that  obtains  when  there 
is  no  flow. 

This  theorem  of  theoretical  hydraulics  is  of  great  importance 
in  practice,  although  it  has  been  deduced  for  mean  veloc- 
ities and  mean  pressure-heads,  while  actually  the  velocity 
and  the  pressure  are  not  the  same  for  all  points  of  the 
cross-section. 

The  pressure-head  at  any  section  hence  decreases 
when  the  velocity  of  the  water  increases.  To  illustrate, 
let  the  depths  of  the  centers  of  ax  and  a2  be  6  and  8  feet 
below  the  water  level,  and  let  their  areas  be  1.2  and  2.4 

*  Hydrodynamica  (Strassburg,  1738),  pp.  35,  144. 


ART.  32  STEADY    FLOW    IN    SMOOTH    PlPES  77 

square  feet.  Let  the  discharge  of  the  pipe  be  14.4  cubic 
feet  per  second.  Then  from  (32)x  the  mean  velocity  in 
av  is  ^  =  14.4/1.2  =  12  feet  per  second,  which  corresponds 
to  a  velocity  head  of  o.oi555^2  =  2.24  feet,  and  conse- 
quently from  (32) 2  the  pressure-head  in  at  is  6.0  —  2.24  = 
3.76.  For  the  section  a2  the  velocity  is  6  feet  per  second 
and  the  velocity  head  is  0.56  feet,  so  that* the  pressure- 
head  is  8.0  —  0.56  =  7.44  feet. 

The  theorem  of  (32) 2  may  be  also  applied  to  the  jet 
issuing  from  the  end  of  the  pipe.  Outside  the  pipe  there 
can  be  no  pressure,  and  if  h  be  the  hydrostatic  head  and 
V  the  velocity  the  equation  gives  h  =  V2/2g,  or  V  =  \/2gh, 
that  is,  if  frictional  resistances  be  not  considered,  the 
theoretic  velocity  of  flow  from  the  end  of  a  pipe  is  that 
due  to  the  hydrostatic  head  upon  it.  In  Chapter  VIII  it 
will  be  seen  that  the  velocity  is  much  smaller  than  this, 
for  a  large  part  of  the  head  h  is  expended  in  overcoming 
friction. 

A  negative  pressure  may  occur  if  the  velocity-head 
becomes  greater  than  the  hydrostatic  head,  for  (32)2 
shows  that  h^  is  negative  when  v^/2g  exceeds  Hlf  A  case 
of  this  kind  is  given  in  Fig.  326,  where  the  section  at  A 
is  so  small  that  the  velocity  is  greater  than  that  due  to 
the  head  Hlt  so  that  if  a  tube  be  inserted  at  A  no  water 
runs  out,  but  if  the  tube  be  carried  down- 
ward into  a  vessel  of  water  there  will 
be  lifted  a  column  CD  whose  height 
is  that  of  the  negative  pressure-head 
hv  For  example,  let  the  cross-section 
of  A  be  0.4  square  feet,  and  its  head  h 
be  4.1  feet,  while  8  cubic  feet  per  second 
are  discharged  from  the  orifice  below.  Then  the  velocity 
at  A  is  20  feet  per  second,  and  the  corresponding  velocity- 
head  is  6.22  feet.  The  pressure-head  at  A  then  is,  from 
the  theorem  of  formula  (32)., 

/^=4. i—  6.22  =  —  2.12  feet 


78  THEORETICAL  HYDRAULICS  CHAP,  m 

and  accordingly  there  exists  at  A  an  inward  pressure 
pl  =  -2.12X0.434=  -0.92  pounds  per  square  inch 

This  negative  pressure  will  sustain  a  column  of  water 
CD  whose  height  is  2.12  feet.  If  the  small  vessel  be 
placed  so  that  its  water  level  is  less  than  2.12  feet  below 
A,  water  will  be  constantly  drawn  from  the  smaller  to 
the  larger  vessel.  This  is  the  principle  of  the  action  of 
the  injector-pump. 

Prob.  32.  In  a  horizontal  tube  there  are  two  sections  of 
diameters  i.o  and  1.5  feet.  The  velocity  in  the  first  section 
is  6.32  feet  per  second,  and  the  pressure-head  is  21.57  feet- 
Find  the  pressure-head  for  the  second  section  if  no  energy  is 
lost  between  the  sections. 

ART.  33.     COMPUTATIONS  IN  METRIC  MEASURES 

(Art.  22)  Using  for  the  acceleration  of  the  mean  value 
9.80  meters  per  second  per  second,  formulas  (22)  2  become 

(33) 


in  which  h  is  in  meters  and  V  in  meters  per  second.  Table 
16  shows  values  of  the  velocity  for  given  heads,  and  values 
of  the  velocity-head  for  given  velocities. 

(Art.  23)  The  area  a  is  in  square  meters,  the  velocity 
V  in  meters  per  second,  and  the  discharge  Q  in  cubic 
meters  per  second.  Thus  if  a  pipe  20  centimeters  in 
diameter  discharges  0.15  cubic  meters  per  second  the 
area  of  the  cross-section  is  0.03142  square  meters  and  the 
mean  velocity  is  0.15/0.03142  =4-77  meters  per  second. 

(Art.  24)  For  Fig.  24  let  the  reservoir  be  one  meter 
in  diameter,  the  load  W  be  2000  kilograms,  and  the  orifices 
be  3  meters  below  the  piston.  ,  Let  the  exterior  head  on 
A  be  1.5  meters,  the  orifice  B  be  open  to  the  atmosphere, 
and  the  orifice  C  be  in  air  whose  pressure  is  0.7  kilograms 
per  square  centimeter.  The  area  of  the  piston  is  0.7854 


ART.  33      COMPUTATIONS  IN  METRIC  MEASURES         79 

square  meters,  and  the  head  corresponding  to  the  pressure 
on  the  upper  water  surface  is 


2000 


Q 

h0=  —  =  -  ^—    ;  -  =  2.546  meters. 
w      0.7854X1000 

The  head  hli&  3  meters  for  the  first  orifice,  o  for  the  second, 
and  —  10(1.033  —  0.7)  =  —3.33  meters  for  the  third.  The 
three  theoretic  velocities  of  outflow  then  are 


V  =  4.427\/3+2.546-i.5  =  8.91  meters  per  second; 
V  =  4.  42  7\/3  +2.546-0  =10.43  meters  per  second, 
V=4.  427^3  +  2.546+3.33  =13.19  meters  per  second. 

If  in  this  example  the  liquid  be  alcohol  which  weighs  800 
kilograms  per  cubic  meter,  the  head  of  alcohol  correspond- 
ing to  the  pressure  of  the  piston  is 

2000 

-3-183  meters- 


and  accordingly  for  discharge  into  the  atmosphere  at  the 
depth  hi  =  3  meters  the  velocity  is 


F  =  4.427V/3+3.i8  =  u.oi  meters  per  second, 
while  for  water  the  velocity  was  10.43  meters  per  second. 

(Art.  28)  As  an  illustration  of  (28)  2  let  water  issue 
from  a  pipe  6  centimeters  in  diameter  with  a  velocity  of 
4  meters  per  second.  The  cross-section  is  found  from 
Table  51  to  be  0.002827  square  meters,  and  then  the 
theoretic  work  in  kilogram-meters  per  second  is 

K  =  o.oio2  X 


which  is  0.123  metric  horse-powers.  If  the  velocity  be 
1  6  meters  per  second  the  stream  will  furnish  7.87  horse- 
powers. 

(Art.  32)     In   Fig.    32a,    suppose   the   sections   at   and 
a2  to  be  0.06  and  0.12  square  meters,  and  the  depths  of 


80  THEORETICAL  HYDRAULICS  CHAP,  in 

their  centers  below  the  water  level  of  the  reservoir  to 
be  4.5  and  5.5  meters.  Let  0.24  cubic  meters  per  second 
be  discharged  from  the  pipe,  then  from  (32)  1  the  mean 
velocities  in  at  and  a2  are  4.0  and  2.0  meters  per  second. 
The  velocity-heads  are  then  0.82  meters  for  ax  and  0.20 
meters  for  a2,  so  that  during  the  flow  the  pressure-head 
at  A  is  4.5  —  0.82  =3.68  meters  and  that  at  B  is  5.5  —  0.20  = 
5.30  meters. 

Prob.  33a.  What  theoretic  velocities  are  produced  by  heads 
of  o.i,  o.oi,  and  o.ooi  meters?  What  is  the  velocity-head  of 
a  jet,  7.5  centimeters  in  diameter,  which  discharges  500  liters 
per  second? 

Prob.  336.  A  prismatic  vessel  has  a  cross-section  of  1.5 
square  meters  and  an  orifice  in  its  base  has  an  area  of  150  square 
centimeters.  Compute  the  theoretic  time  for  the  water  level 
to  drop  3  meters  when  the  head  at  the  beginning  is  4  meters. 

Prob.  33c.  A  small  turbine  wheel  using  3  cubic  meters  of 
water  per  second  under  a  head  of  ioj  meters  is  found  to  deliver 
5.1  metric  horse-powers.  Compute  the  efficiency  of  the  wheel. 

Prob.  33 d.  In  an  inclined  tube  there  are  two  sections  of 
diameters  10  and  20  centimeters,  the  second  section  being 
1.536  meters  higher  than  the  first.  The  velocity  in  the  first 
section  is  6  meters  per  second  and  the  pressure-head  is  7.045 
meters.  Find  the  pressure-head  for  the  second  section. 


ART.  34  GENERAL    CONSIDERATIONS  81 


CHAPTER  IV 
INSTRUMENTS  AND  OBSERVATIONS 

ART.  34.     GENERAL  CONSIDERATIONS 

Some  of  the  most  important  practical  problems  of 
Hydraulics  are  those  involving  the  measurement  of  the 
amount  of  water  discharged  in  one  second  from  an  orifice, 
pipe,  or  conduit  under  given  conditions.  The  theoretic 
formulas  of  the  last  chapter  furnish  the  basis  of  most  of 
these  methods,  and  in  the  chapters  following  this  one 
are  given  coefficients  derived  from  experience  which 
enable  those  formulas  to  be  applied  to  practical  conditions. 
These  coefficients  have  been  determined  by  measuring 
heads,  pressures,  or  velocities  with  certain  instruments, 
and  also  the  amount  of  water  actually  discharged,  and 
then  comparing  the  theoretic  results  with  the  actual 
ones.  It  is  the  main  object  of  this  chapter  to  describe 
the  instruments  used  for  this  purpose,  and  a  few  remarks 
concerning  advantageous  methods  for  the  discussion  of  the 
observations  will  also  be  made. 

The   engineer's   steel  tape,    level,    and   transit   are  in- 
dispensable tools  in  many  practical  hydraulic  problems. 
For  example,  two  reservoirs  M        M 
*and    Ar,    connected    by    a    pipe 
line,  may  be  several  miles  apart. 
To   ascertain   the   difference   in 
elevation    of    their    water    sur- 
faces lines  of  levels  may  be  run 
and    bench    marks    established 
near  each  reservoir  as  also  at  other  points  along  the  pipe 


82  INSTRUMENTS  AND  OBSERVATIONS  CHAP,  iv 

line.  From  the  bench  marks  at  the  reservoirs  there  can 
be  set  up  simple  board  gages,  so  that  simultaneous  read- 
ings can  be  taken  at  any  time  to  find  the  difference  in 
elevation.  From  the  bench  marks  along  the  pipe  line  a 
profile  of  the  same  can  be  plotted  for  use  in  the  discussion. 
With  the  transit  and  tape  the  alignment  of  the  pipe  line 
and  the  lengths  of  its  curves  and  tangents  can  also  be  taken 
and  mapped.  All  of  these  records,  in  fact,  are  necessary 
in  order  to  determine  the  amount  of  water  delivered 
through  the  pipe. 

For  work  on  a  smaller  scale,  like  that  of  the  discharge 
from  an  orifice  in  a  tank,  the  steel  tape  may  be  used  to 
mark  points  from  which  a  glass  gage  tube 
may  be  set  upon  which  the  height  of  the 
water    surface    above    the    orifice    can    be 
read  at   any  time   during  the   experiment. 
Another  method  is  to  have  a  float  on  the 
water  surface,  the  vertical  motion  of  which 
Fi    346        *s  communicated  to  a  cord  passing  over  a 
pulley,  so  that  readings  can  be  taken  on  a 
scale  as  the  weight  at  the  lower  end  of  the  cord    moves 
up    or    down.     When    the  head  is  very  small,   however, 
these  methods  are  not  sufficiently  precise  and  the  hook 
gage,  described  in  Art.  35,  must  be  used. 

A  small  quantity  of  water  flowing  from  an  orifice  may 
be  measured  by  allowing  it  to  run  into  a  barrel  set  upon 
a  platform  weighing  scale.  The  weight  of  water  dis- 
charged in'  a  given  time  is  thus  ascertained,  the  time  being 
noted  by  a  stop-watch,  and  the  volume  is  then  computed 
by  the  help  of  Table  7.  If  the  flow  is  uniform  the  dis- 
charge in  one  second  is  then  found  by  dividing  the  volume 
by  the  number  of  seconds.  A  larger  quantity  of  water 
may  be  measured  in  a  rectangular  tank,  the  cross-section 
of  which  is  accurately  known ;  here  the  water  surface  is 
noted  at  the  beginning  and  end  of  the  experiment,  and 
the  volume  is  then  computed  by  multiplying  the  area 


ART.  34 


GENERAL  CONSIDERATIONS 


83 


by  the  difference  of  the  two  elevations.  For  example, 
if  a  square  tank  be  4  feet  2  inches  inside  dimensions,  and 
if  the  gage  reads  3.17  feet  at  the  beginning  and  4.62  feet 
at  the  end  of  the  experiment,  which  lasted  304  seconds, 
the  flow,  if  uniform,  is  0.0828  cubic  feet  per  second. 

Larger  quantities  of  water  still  are  sometimes  measured 
in  the  reservoir  of  a  city  supply.  The  engineer,  by  the 
use  of  his  level,  transit,  and  tape,  makes  a  precise  contour 
map  of  the  reservoir,  determines  with  the  planimeter 
the  area  enclosed  by  each  contour  curve,  and  computes 
the  volume  included  between  successive  contour  planes. 
For  instance,  if  the  area  of 
the  contour  curve  AB  be 
84  320  square  feet  and  that 
of  CD  be  79  624  square 
feet  and  the  vertical  dis- 
tance between  the  contour 
planes  be  5  feet,  the  volume 
included  is  409  860  cubic 
feet  by  the  method  of 
mean  areas.  A  more  pre- 
cise determination,  how- 
ever, may  be  made  by  measuring  the  area  of  a  contour 
curve  half-way  between  A B  and  AC;  if  this  be  found  to 
be  82  150  square  feet,  the  volume  included  between  AB 
and  AC  is  computed  by  the  prismoidal  formula  and  found 
to  be  410  450  cubic  feet. 

These  direct  methods  of  water  measurement  form  the 
basis  of  all  hydraulic  practice.  In  this  manner  water 
meters  are  rated,  and  the  coefficient  determined  by  which 
practical  formulas  for  flow  through  orifices,  weirs  and 
pipes  are  established.  These  coefficients  being  known, 
indirect  methods  may  be  used  for  water  measurement, 
namely,  the  discharge  can  be  computed  from  the  formulas 
after  area  and  heads  have  been  ascertained.  There  are 
also  methods  of  indirect  measurement  from  observed 


FIG.  34<; 


84 


INSTRUMENTS  AND  OBSERVATIONS 


CHAP.  IV 


velocities  which  will  be  described  later,  and  which  are 
especially  valuable  in  finding  the  discharge  of  pipes,  con- 
duits and  streams. 

Prob.  34.  Water  flows  from  an  orifice  uniformly  for  93.5 
seconds  and  falls  into  a  barrel  on  a  platform  weighing  scale. 
The  weight  of  the  empty  barrel  is  28  pounds  and  that  of  the 
barrel  and  water  is  267  pounds.  What  is  the  discharge  of  the 
orifice  in  gallons  per  minute,  if  the  temperature  of  the  water 
is  52°  Fahrenheit? 

ART.  35.     THE  HOOK  GAGE 

The  hook  gage,  invented  by  Boyden  about  1840,  con- 
sists of  a  graduated  metallic  rod  sliding  vertically  in  fixed 
supports,  upon  which  is  a  vernier  by  which  readings  can  be 
taken  to  thousandths  of  a  foot.  At  the  lower 
end  of  the  rod  is  a  sharp-pointed  hook,  which  is 
raised  or  lowered  until  its  point  is  at  the  water 
level.  Fig.  35a  represents  the  form  of  hook  gage 
made  by  Gurley,  the  graduation  on  the  rod  being 
to  feet  and  hundredths.  The  graduation  has  a 
length  of  2.2  feet,  so  that  variations  in  the  water 
level  of  less  than  this  amount  can  be  measured, 
by  using  the  vernier,  to  thousandths  of  a  foot. 
To  take  a  reading  on  a  water  surface,  the  point 
of  the  hook  is  lowered  below  the  surface  and  then 
slowly  raised  by  the  screw  at  the  top  of  the  in- 
strument. Just  before  the  point  of  the  hook 
pierces  the  skin  of  the  water  (Art.  3)  a  pimple 
or  protuberance  is  seen  to  rise  above  it ;  the  hook 
is  then  depressed  until  the  pimple  is  barely  visi- 
ble and  the  vernier  is  read.  The  most  precise 
hook  gages  read  to  ten-thousandths  of  a  foot,  arid 
it  has  been  stated  that  an  experienced  observer 
can,  in  a  favorable  light  and  on  a  water  surface 
perfectly  quiet,  detect  differences  of  level  as 
FIG.  35a  small  a§  0.0002  feet. 
A  cheaper  form  of  hook  gage,  and  one  sufficiently  pre- 


ART.  35  THE  HOOK  GAGE  85 

else  in  some  classes  of  work,  can  be  made  by  screwing  a 
hook  into  the  foot  of  an  engineer's  leveling  rod.  The  back 
part  of  the  rod  is  then  held  in  a  vertical  position  by  two 
clamps  on  fixed  supports,  while  the  front  part  is  free  to 
slide.  It  is  easy  to  arrange  a  slow-motion  movement  so 
that  the  point  of  the  hook  may  be  precisely  placed  at  the 
water  level.  The  reading  of  the  vernier  is  determined  when 
the  point  of  the  hook  is  at  a  known  elevation  above  an  ori- 
fice or  the  crest  of  a  weir,  and  by  subtracting  from  this  the 
subsequent  readings  the  heads  of  water  are  known.  A 
New  York  rod,  reading  to  thousandths  of  a  foot,  is  to  be 
preferred. 

Hook  gages  are  principally  used  for  determining  the 
elevations  of  the  water  surface  above  the  crest  of  a  weir, 
as  the  heads  of  water  are  small  and  must 
be  known  with  precision.  In  Fig.  35b, 
the  crest  of  the  weir  is  seen  and  the  hook 
gage  is  erected  at  some  distance  back 
from  it,  where  the  water  surface  is 
level.  In  this  case  great  care  should 
be  taken  to  determine  the  reading  cor- 
responding to  the  level  of  the  crest.  In  the  larger  forms  of 
hooks  this  may  be  done  by  taking  elevations  of  the  crest 
and  of  the  point  of  the  hook  by  means  of  an  engineer 's  level 
and  a  light  rod.  With  smaller  hooks  it  may  be  done  by 
having  a  stiff  permanent  hook,  the  elevation  of  whose  point 
with  respect  to  the  crest  is  determined  by  precise  levels; 
the  water  is  then  allowed  to  rise  slowly  until  it  reaches  the 
point  of  this  stiff  hook,  when  readings  of  the  vernier  of  the 
lighter  hook  are  taken.  Another  method  is  to  allow  a  small 
depth  of  water  to  flow  over  the  crest  and  to  take  readings 
of  the  hook,  while  at  the  same  time  the  depth  on  the  crest 
is  measured  by  a  finely  graduated  scale.  Still  another  way 
is  to  allow  the  water  to  rise  slowly,  and  to  set  the  hook  at 
the  water  level  when  the  first  filaments  pass  over  the  crest ; 
this  method  is  not  a  very  precise  one  on  account  of  capillary 


86  INSTRUMENTS  AND  OBSERVATIONS  CHAP,  iv 

attraction  along  the  crest.  As  the  error  in  setting  the 
hook  is  a  constant  one  which  affects  all  the  subsequent 
observations,  especial  care  should  be  taken  to  reduce  it  to 
a  minimum  by  taking  a  number  of  observations  in  order 
to  obtain  a  precise  mean  result. 

The  hook  gage  is  also  used  to  find  the  difference  of  the 
water  levels  in  tanks  for  experiments  for  the  determination 
of  hydraulic  coefficients,  and  in  wells  along  pipe  lines  when 
experiments  are  made  to  investigate  frictional  resistances. 
In  general  its  use  is  confined  to  cases  where  the  head  is 
small,  as  for  high  heads  so  great  a  degree  of  precision  is 
not  required  (Art.  55). 

Prob.  35.  A  wooden  tank,  4.52  by  5.78  feet  in  inside  dimen- 
sions, has  leakage  near  its  base.  The  hook  gage  reads  2.047  feet 
at  11.57  A.M.,  1.470  feet  at  12.05  p-M->  and  0.938  feet  at  12.13  P.M. 
Show  that  the  probable  leakage  in  the  first  and  last  minutes  was 
1.96  and  1.66  cubic  feet. 

ART.  36.     PRESSURE  GAGES 

A  pressure  gage,  often  called  a  piezometer,  is  an  instru- 
ment for  measuring  the  pressure  of  water  in  a  pipe.  The 
form  most  commonly  found  in  the  market  has  a  dial  and 
movable  pointer,  the  dial  being  graduated  to  read  pounds 
per  square  inch.  The  principle  on  which  this  gage  acts 
is  the  same  as  that  of  the  Richard  aneroid  barometer  and 
the  Bourdon  steam  gage.  Within  the  case  is  a  small  coiled 
tube  closed  at  one  end,  while  the  other  end  is  attached  to 
the  opening  through  which  the  water  is  admitted.  This 
tube  has  a  tendency  to  straighten  when  under  pressure 
and  thus  its  closed  end  moves  and  the  motion  is  communi- 
cated to  the  pointer ;  when  the  pressure  is  relieved  the  tube 
assumes  its  original  position  and  the  pointer  returns  to  zero. 
There  is  no  theoretical  method  of  determining  the  motion 
of  the  pointer  due  to  a  given  pressure,  and  this  is  done  by 
tests  in  which  known  pressures  are  employed,  and  accord- 


ART.  36 


PRESSURE  GAGES 


87 


ingly  the  divisions  on  the  graduated  scale  are  usually  un- 
equal. These  gages  are  liable  to  error  after  having  been 
in  use  for  some  time,  especially  so  at  high  pressures,  and 
hence  should  be  tested  before  and  after  any  important 
-series  of  experiments. 

In  most  hydraulic  work  the  head  of  water  causing  the 
pressure  is  required  to  be  known.  If  p  be  the  gage  reading 
in  pounds  per  square  inch  the  head  of  water  in  feet  is 
A  =  2. 304/7,  or  if  p  be  the  gage  reading  in  kilograms  per 
square  centimeter,  the  head  of  water  in  meters  is  h  =  lop. 
The  graduation  of  the  gage  dial  may  be  made  to  read  heads 
directly,  so  as  to  avoid  the  necessity  of  reduction. 

The  pressure  at  any  point  of  a  pipe  may  be  measured 
"by  the  height  of  a  column  of  water  in  an  open  tube,  as  seen 
at  A  in  Fig.  36a.  The  upper 
portion  of  the  tube  may  be  of 
glass,  so  that  the  position  of 
the  water  level  may  be  noted 
on  a  scale  held  alongside.  It  is 
not  necessary  that  the  water 
column  should  be  vertical,  and  = 
a  hose  is  often  used,  as  seen  at  FIG.  36a 

B,  with  a  glass  tube  at  its  top.  At  C  is  shown  a  dial  pres- 
sure gage.  When  the  head  h  is  directly  read  in  feet,  the 
pressure  in  pounds  per  square  inch  may  be  computed  from 
^  =  0.434/2.  In  order  to  secure  precise  results  when  the 
water  in  the  pipe  is  in  motion,  it  is  necessary  that  a  piez- 
ometer tube  be  inserted  into  the  pipe  at  right  angles;  if 
inclined  toward  or  against  the  current  the  head  h  is  greater 
or  less  than  that  due  to  the  actual  pressure  at  its  mouth. 

For  high  pressures  a  water  column  is  impracticable  on 
account  of  its  great  height,  and  hence  mercury  gages  are 
used.  Fig.  366  shows  the  principle  of  construction,  a  bent 
tube  ABC  with  both  ends  open,  having  mercury  in  its  lower 
portion,  and  the  water  column  of  height  h  being  balanced 


88 


INSTRUMENTS  AND  OBSERVATIONS 


CHAP.  IV 


by  the  mercury  column  of  height  z.      If  the  atmospheric- 
pressures  at  A  and  C  are  the  same,  it  is  evident, 
Lri  from  Art.  5,  that  the  height  h  is  about   13.6 

times  the  height  0,  since  the  specific  gravity  of 
mercury  is  about  13.6.  Now  z  can  be  read  on 
a  scale  placed  between  the  legs  of  the  tube, 
and  thus  h'is  known,  as  also  the  water  pressure 
at  the  point  B.  If  the  atmospheric  pressures 
t  at  A  and  C  are  different,  as  will  be  the  case  when 

h  is  very  large,  let  b±  be  the  barometer  reading 
at  A  and  b2  that  at  C,  both  being  in  the  same 
linear  unit  as  h  and  z.     The  absolute  pressure 
at  B  is  that  due  to  the  height  sh  +  sfbl1  where 
•  5  and  sf  are  the  specific  gravities  of  water  and 
mercury,  and  the  absolute  pressure  at  the  same 
elevation  in  the  other  leg  is  that  due  to  the  height 
FIG.  36b       s*(z  +  b2).     Since  these  pressures  are  equal, 

h  =  (sf/s)  (z  +  b2  —  b^ 

is  the  head  corresponding  to  the  distance  z  on  the  scale.. 
The  ratio  s'/s  is  13.6  approximately,  its  actual  value  de- 
pending on  the  purity  of  the  water  and  mercury  and  on  the 
temperature. 

Fig.  36c    shows    the   mercury    gage   as   arranged   for 
measuring  the  pressure-head  at  a  point  A  in  a  water  pipe. 
The  top  is  open  to  the  air  and  through  it 
the  mercury  may  be  poured  in,  the  cock 
E  being  closed  and  F  open;  the  mercury 
then  stands  at  the  same  height  in  each 
tube.     The   cock   F  being   closed   and   E 
opened,  the  water  enters  the  left-hand  tube, 
depressing  the  mercury  to   B,   causing  it 
to  rise  to  C  on  the  other  side.     The  dis- 
tance z  is  then  read  on  a  scale  between 
the  two  tubes,  and  the  height  of  B  above       ~~ 
A  by  another  scale.     The  pressure  of  the 
water  at  B  is  that  due  to  the  head  13.62,  and  the  pressure 


ART.  36 


PRESSURE  GAGES 


89 


at  A  is  that  due  to  the  head  y  +  i$.6z.  In  precise  work 
it  is  necessary  to  determine  the  exact  specific  gravity  of 
the  mercury  and  water  at  different  temperatures,  so  that 
precise  values  of  the  ratio  sf/s  may  be  known.  The  value 
of  s'  depends  upon  the  purity  of  the  mercury  and  is  some- 
times lower  than  13.56. 

For  very  high  pressures,  such  as  are  used  in  operating 
heavy  forging-presses,  the  mercury  column  of  the  above 
gage  would  be  so  long  as  to  render  it  impracticable,  and 
accordingly  other  methods  must  be  employed.  Fig.  36d 
represents  a  mercury  gage  constructed  on  the  principle 
of  the  hydraulic  press  (Art.  10).  W  is  a  small  cylinder 
into  which  the  water  is  admitted  through  the  small  pipe 
at  the  top,  and  M  is  a  large  cylinder  containing  mercury 
to  which  a  glass  tube  is  attached. 
Before  the  water  is  admitted 
into  W  the  mercury  stands  at 
the  level  of  B  in  both  the  glass 
tube  and  large  cylinder,  if  the 
piston  does  not  rest  on  the 
mercury.  When  the  water  is 
admitted  its  pressure  on  the 
upper  end  of  the  piston  is  pa, 
if  p  be  the  unit-pressure  and  a 
the  area  of  the  upper  end.  If  A  be  the  area  of  the  lower 
end  of  the  piston  the  total  pressure  upon  it  is  also  pa, 
and  hence  the  unit-pressure  on  the  mercury  surface  is 
p.  a  I 'A,  and  this  is  balanced  by  the  column  of  height  z 
in  the  glass  tube.  For  example,  suppose  that  A  =  2ooa, 
then  the  unit-pressure  on  the  mercury  surface  is  o.oo$p; 
further,  if  z  be  60  inches,  the  unit-pressure  at  B  is  about 
2X14.7=29.4  pounds  per  square  inch  (Art.  5),  and  ac- 
cordingly the  pressure  in  W  is  £  =  200X29.4  =  5880 
pounds  per  square  inch,  which  corresponds  to  a  head  of 
water  of  about  13  550  feet. 


FIG.  3Qd 


90 


INSTRUMENTS  AND  OBSERVATIONS 


CHAP.  IV 


Prob.  36.  The  diameter  of  the  large  end  of  the  piston  in  the 
last  figure  is  15  inches,  and  the  diameter  of  the  mercury  column 
is  J  inch.  Find  the  distance  the  piston  is  depressed  when  the 
mercury  rises  60  inches.  What  is  the  total  pressure  on  the 
piston  ? 

ART.  37.     DIFFERENTIAL  PRESSURE  GAGES 

A  differential  gage  is  an  instrument  for  measuring 
differences  of  heads  or  pressures,  and  this  must  be  frequent- 
ly done  in  hydraulic  work.  One  of  the  simplest  forms 
is  that  seen  in  Fig.  37a  where  two 
water  columns  from  A  and  D  are 
brought  to  the  sides  of  a  common 
scale'  upon  which  the  difference  of 
height  EC  is  directly  read.  A  better 
form  is  one  having  two  glass  tubes 
fastened  to  a  scale,  these  tubes  being 
provided  with  attachments  upon  which 
cari  be  screwed  the  hose  leading 
from  the  pipe.  With  these  forms,  however,  large  heads 
cannot  be  managed  even  if  their  difference  be  small,  and 
hence  the  mercury  gage  was  devised. 

Fig.  376  shows  the  principle  of  the  mercury  differential 
gage.*  Two  parallel  tubes  are  open  at  the  top,  and  here 
the  mercury  is  poured  in,  the  cocks  E  and 
F  being  open  and  A  and  C  closed;  the  mer- 
cury then  stands  at  the  same  height  in  each 
tube.  The  cocks  E  and  F  being  now  closed 
and  A  and  C  opened,  the  water  enters  at 
A  and  C,  and  the  mercury  is  depressed  in 
one  tube  and  elevated  in  the  other.  Let 
the  pressure  at  E  be  that  due  to  the  head 
hv  and  the  pressure  at  C  be  that  due  to  the 
head  h2,  and  let  h^  be  greater  than  h2 ;  also  let  the  distance 


FIG.  37a 


E 


FIG.  376 


*  For  the    details  of  construction  see  paper  by  Kuichling  in  Transac- 
tions American  Society  of  Civil  Engineers,  1892,  vol.  26,  p.  439. 


ART.  37  DIFFERENTIAL   PRESSURE    GAGES  91 

read  on  the  scale  between  the  two  tubes  be  z.  Then 
7^=  ^2  +  13. 60,  or  the  difference  of  the  heads  of  water  on 
B  and  C  is  h1  —  h2  =  13.60.  Thus  if  z  be  1.405  feet,  the 
difference  of  the  heads  is  19.1  feet.  Here,  as  for  the  mer- 
cury gage  of  Art.  36,  the  specific  gravity  of  the  mercury 
and  water  must  be  known  for  different  temperatures, 
or  comparisons  of  the  instrument  with  a  standard  gage 
must  be  made. 

When  the  difference  of  the  heads  is  small  the  water 
gage,  explained  in  the  first  paragraph,  cannot  measure 
it  with  precision,  especially  when  the  columns  are  subject 
to  oscillations.  To  increase  the  distance  between  B  and 
C  and  at  the  same  time  decrease  the  amount  of  oscillation, 
the  oil  differential  gage,  invented  by  Flad  in  1885,  may 
be  used.  Fig.  37 c  shows  the  principle  of  construction.* 
The  cocks  A  and  D  being  closed  and  F 
open,  sufficient  oil  is  poured  in  at  F  to 
partially  fill  the  two  tubes.  Then  F  is 
closed  and  the  water  admitted  at  A  and 
D,  when  it  rises  to  B  in  one  tube  and  to 
C  in  .  the  other,  the  oil  filling  the  tubes 
above  the  water.  Let  5  be  the  specific 
gravity  of  the  water  and  s'  that  of  the 
oil,1  let  hi  be  the  head  of  water  on  B  and  Fio.  37c 
/z2  that  on  C,  then  sh2=skl  —  sfz,  whence  h2—hl=(s'/s)z. 
Kerosene  oil  having  a  specific  gravity  of  about  0.79  is  gen- 
erally used,  and  if  the  specific  gravity  of  the  water  be  unity, 
the  difference  of  the  heads  is  0.792.  Thus  z  is  greater 
than  h2-hlJ  and  hence  an  error  in  reading  z  produces  a 
smaller  error  in  h2  —  hl.  The  specific  gravities  of  the  oil 
and  water  must  be  determined,  however,  so  that  s?/s 
can  be  expressed  to  four  significant  figures  when  precise 
work  on  low  heads  is  to  be  done. 

*  For  the  details  see  paper  by  Williams,  Hubbell,  and  Fenkell  in 
Transactions  of  American  Society  of  Civil  Engineers,  1902,  voL  47, 
pp.  72— 83. 


92 


INSTRUMENTS  AND  OBSERVATIONS 


CHAP.  IV 


The  difference  of  head  ht  —  h2l  determined  by  these 
differential  gages,  is  the  difference  of  the  heads  due  to 
the  pressure  at  the  water  levels  B  and  C.  The  difference 
of  the  actual  heads  at  the  points  of  connection  with  the 
pipe  under  test  is  next  to  be  determined.  Fig.  37 'd  shows 
a  mercury  gage  set  over  a  water  pipe  for  the  purpose  of 
determining  the  loss  of  head  due  to  a  valve,  the  velocity 


FIG.  37d 


FIG.  37e 


of  the  water  being  high,  so  that  the.  difference  of  pressure 
at  A  and  D  is  large.  Fig.  37  'e  shows  an  oil  gage  set  over 
a  similar  pipe,  the  velocity  being  low,  so  that  the  difference 
of  pressure  is  small.  Let  a  horizontal  plane,  represented 
by  the  broken  line,  be  drawn  through  the  zero  of  the  scale 
of  the  gage,  and  let  d  be  the  distance  of  this  plane  above 
the  horizontal  pipe.  Let  b  and  c  be  the  readings  of  this 
scale  at  the  water  levels  B.  and  C  in  the  gage  tubes,  the 
difference  of  these  readings  being  z.  Let  hl  and  h2  be  the 
pressure-heads  on  B  and  C,  and  H1  and  H2  those  on  A 
and  D.  Then  Hl=h1  +  b  +  d  and  H2=h2  +  c  +  d,  and  the 
difference  of  these  heads  is 


which  is  applicable  to  both  kinds  of  differential  gages. 
For  the  mercury  gage  the  head/^  —  h2  equals  13.62,  while 
the  value  of  b  —  c  is  —  z  ;  hence 


I  —  H2  =  13. 60  —  2  =  12. 6z 


ART.  38  WATER    METERS  93 

For  the  oil  gage  h1—  h2  is  —  0.792,   while  b  —  c  is  z,  hence 


In  general,  if  sf  be  the  ratio  of  the  specific  gravity  of  the 
mercury  or  oil  to  that  of  the  water,  the  difference  of  the 
pressure-heads  at  A  and  D  which  is  the  loss  of  head  due 
to  the  valve,  is  (5'  —  1)2  for  the  mercury  gage  and  (  i—s')z 
for  the  oil  gage. 

The  principle  of  the  mercury  gage  can  also  be  applied 
to  the  measurement  of  small  differences  of  head  by  using 
a  liquid  having  a  specific  gravity  but  little  heavier  than 
water.  Thus  Cole,  in  1897,*  employed  a  mixture  of  carbpn 
tetrachloride  and  gasoline  which  had  a  specific  gravity  of 
1.25;  for  this  mixture  H1  —  H2  equals'  0.252,  or  2  is  four 
times  the  head  H^  —  H2y  and  accordingly  when  H1  —  H2  is 
small  the  error  in  determining  it  by  the  reading  z  is  greatly 
diminished.  It  may  be  also  noted  that  when  the  tube  or 
pipe  is  not  horizontal  the  expressions  (s'  —  i)z  and  (i—s')z 
give  the  loss  of  head  between  the  two  points  A  and  D, 
although  the  difference  of  the  actual  pressure-heads  may 
be  greater  or  less  according  as  A  is  lower  or  higher  than 
D  (Art.  82). 

Prob.  37.  In  the  case  of  Fig.  37  d  let  the  point  D  be  lower 
than  A  by  0.45  feet,  and  let  the  reading  z  be  0.127  feet-  Show 
that  the  pressure-head  at  A  is  1.15  feet  greater  than  that  at  D. 

ART.  38.     WATER  METERS 

Meters  used  for  measuring  the  quantity  of  water  sup- 
plied to  a  house  or  factory  are  of  the  displacement  type, 
that  is,  as  the  water  passes  through  the  meter  it  displaces 
or  moves  a  piston,  a  wheel,  or  a  valve,  the  motion  of  which 
is  communicated  through  a  train  of  clock  wheels  to  dials 
where  the  quantity  that  has  passed  since  a  certain  time 

*  Transactions  American  Society  Civil  Engineers,  1902,  vol.  47,  p.  276. 


94  INSTRUMENTS  AND  OBSERVATIONS  CHAP,  iv 

is  registered.  There  is  no  theoretical  way  of  determining 
whether  or  not  the  readings  of  the  dial  hands  are  correct, 
but  each  meter  must  he  rated  by  measuring  the  discharge 
in  a'  tank.  Several  meters  may  be  placed  on  the  same  pipe 
line  in  this  operation,  the  same  discharge  then  passing 
through  each  of  them.  When  impure  water  passes  through 
a  meter  for  any  length  of  time  deposits  are  liable  to  impair 
the  accuracy  of  its  readings,  and  hence  it  should  be  rerated 
at  intervals. 

The  piston  meter  is  one  in  which  the  motion  of  the  water 
causes  two  pistons  to  move  in  opposite  directions,  the  water 
leaving  and  entering  the  cylinder  by  ports  which  are  opened 
and  closed  by  slide  valves  somewhat  similar  to  those  used 
in  the  steam-engine.  The  rotary  meter  has  a  wheel  en- 
closed in  a  case  so  that  it  is  caused  to  revolve  as  the  water 
passes  through.  The  screw  meter  has  an  encased  helical 
surface  that  revolves  on  its  axis  as  the  water  enters  at  one 
end  and  passes  out  at  the  other.  The  disk  meter  has  a 
wabbling  disk  so  arranged  that  its  motion  is  communicated 
to  a  pin  which  moves  in  a  circle.  In  all  these,  and  in  many 
other  forms,  it  is  intended  that  the  motion  given  to  the 
pointers  on  the  dials  shall  be  proportional  to  the  volume  of 
water  passing  through  the  meter.  The  dials  may  be  arranged 
to  read  either  cubic  feet  or  gallons,  as  may  be  required  by 
the  consumers.  These  meters  are  of  different  sizes,  accord- 
ing to  the  quantity  of  water  required  to  be  registered,  and 
the  capacity  of  the  largest  size  is  about  200  cubic  feet  per 
minute. 

The  Venturi  meter,  named  after  the  distinguished 
hydraulician  who  first  experimented  on  the  principle  by 
which  it  operates,  was  invented  by  Herschel  in  1887.* 
Fig.  38a  shows  a  horizontal  pipe  having  an  area  ax  at  each 
end,  and  the  central  part  contracted  to  the  area  a2,  with 
two  small  piezometer  tubes  into  which  the  water  rises. 

*  Transactions  American  Society  Civil  Engineers,  1887,  vol.  17,  p.  228, 


ART.  38 


WATER  METERS 


95 


When  there  is  no  flow  the  water  stands  at  the  same  level 

in  these  two  columns,  but  when  it  is  in  motion  the  heights  of 

these  columns  above 

the  axis  of  the  pipe 

are  hl  and  H2.     Let  vt 

and  v2  be  the   mean 

velocities  in  the  two 

cross-sections.    Then 

by  Art.  25  the  effect-  FlG-  38a 

ive  head  in  the  upper  section  is  h1  +  vl2/2g,  and  that  in 

the  small  section  is  h2  +  v22/2g',   if  there  be  no  losses  caused 

by    friction   these    two    expressions    must   be  equal,  and 

hence  by  the  theorem  of  (32)2, 


Now  let  Q  be  the  discharge  through  the  pipe,  or  Q  =  alvl 
and  also  Q  =  a2v2.  Taking  the  values  of  vt  and  v2  from  these 
expressions,  inserting  them  in  the  above  equation,  and 
solving  for  Q  gives 


(38) 


which  may  be  called  the  theoretic  discharge.  Owing  to 
frictional  losses  which  occur  between  the  two  cross-sections 
the  actual  discharge  q  is  always  less  than  Q,  or  q  =  cQ,  in 
which  c  is  a  coefficient  whose  value  generally  lies  between 
0.95  and  0.99.  To  determine  q,  when  the  coefficient  is 
known,  it  is  hence  only  necessary  to  measure  the  difference 
hi  —  h2l  and  then  compute  Q  by  formula  (38). 

The  Venturi  meter  is  used  for  measuring  the  discharge 
through  water  mains  of  six  inches  or  more  in  diameter. 
The  contracted  area  is  usually  one-ninth  of  the  area  of  the 
pipe  and  hence  the  velocity  through  it  is  nine  times  that 
in  the  pipe.  The  two  columns  of  water  in  practice  are  led 
to  a  mercury  gage  where  the  difference  of  head  h^  —  k^  is 
shown  by  the  difference  in  level  of  the  two  mercury  col- 


96  INSTRUMENTS  AND  OBSERVATIONS  CHAP,  iv 

umns  (Art.  36).  A  scale  with  unequal  divisions  enables 
the  discharge  q  to  be  read  at  a  glance,  and  a  continuous 
record  of  the  same  is  also  made  by  an  automatic  register- 
ing device.  This  meter  is  extensively  used  for  measuring 
the  quantity  of  water  flowing  from  a  reservoir,  or  that 
delivered  to  a  town,  and  its  capacity  is  far  greater  than 
that  of  any  other  form  yet  devised. 

All  meters  cause  a  loss  in  pressure,  so  that  the  pressure- 
head  in  the  pipe  beyond  the  meter  is  less  than  that  in  the 
pipe  as  it  enters  the  meter.  This  is  due  to  the  energy  lost 
in  overcoming  friction.  For  a  Venturi  meter  of  the  pro- 
portions indicated  above  the  loss  of  head  in  feet  is  about 
0.002 1?;2,  where  v  is  the  velocity  in  the  contracted  section 
in  feet  per  second.  Thus,  if  the  velocity  in  a  water  main 
be  3  feet  per  second,  the  velocity  in  the  contracted  section 
will  be  27  feet  per  second,  and  the  loss  of  pressure-head 
due  to  the  meter  is  about  1.53  feet. 

Another  method  of  gaging  the  flow  of  a  pipe  is  by  means 
of  the  Pitot  tube  (Art.  41)  and  a  differential  gage,  whereby 
the  velocity  is  determined  by  measurement  of  a  head  of 
water.  This  apparatus  is  called  the  pitometer,  and  it  has 
the  advantage  that  little  or  no  loss  of  head  results  from  the 
introduction  of  the  tube  into  the  pipe,  but  careful  rating 
is  necessary  in  order  that  recorded  discharges  may  be 
correct. 

Prob.  38.  A  1 2-inch  pipe  delivers  810  gallons  per  minute 
through  a  Venturi  meter.  Compute  the  mean  velocities  in 
the  sections  at  and  a2.  If  the  pressure-head  in  at  is  21.4  feet 
compute  the  pressure-head  in  a2. 

ART.  39.     MEASUREMENT  OF  VELOCITY 

In  Chapter  III  the  velocity  of  flow  from  an  orifice,  or 
in  a  tube  or  pipe,  was  regarded  as  uniform  over  the  cross- 
section.  If  a  be  that  area,  and  v  the  uniform  velocity,  the 
discharge  is  q  =  av;  hence,  if  a  and  v  can  be  found  by  meas- 


ART.  39  MEASUREMENT   OF  VELOCITY  97 

urement  q  is  known.  In  fact,  however,  the  velocity  varies 
in  different  parts  of  a  cross-section,  so  that  the  determin- 
ation of  v  cannot  be  directly  made.  Yet  there  always  is 
a  certain  value  for  v,  which  multiplied  into  a  will  give  the 
actual  discharge  g,  and  this  value  is  called  the  mean  velocity. 

In  the  case  of  a  stream  or  open  channel  the  velocity  is 
much  less  along  the  sides  and  bottom  than  near  the  middle. 
A  rough  determination  of  the  mean  velocity  may  be  made, 
however,  by  observing  the  greatest  surface  velocity  by  a 
float,  and  taking  eight-tenths  of  this  for  the  approximate 
mean  velocity.  Thus,  if  the  float  requires  50  seconds  to 
run  120  feet,  the  mean  velocity  is  about  1.9  feet  per  second; 
then  if  the  cross-section  of  the  stream  be  820  square  feet 
the  approximate  discharge  is  1560  cubic  feet  per  second. 

The  practical  object  of  determining  the  mean  velocity 
is,  in  nearly  all  cases,  to  determine  the  discharge,  but  as 
a  rule  the  mean  velocity  cannot  be  directly  observed. 
A  knowledge  of  its  value,  however,  is  necessary  in  all 
branches  of  hydraulics,  since  hydraulic  coefficients  and 
formulas  are  based  upon  it.  Accordingly,  many  experi- 
ments have  been  made  upon  small  orifices  and  pipes  by 
catching  the  flow  in  tanks  and  thus  determining  q,  then 
the  mean  velocity  has  been  computed  from  v=q/a.  This 
process  has  been  extended,  by  indirect  methods,  to  large 
orifices  and  pipes,  and  finally  to  canals  and  rivers. 

A  common  method  of  finding  the  discharge  of  a  stream 
is  to  subdivide  the  cross-section  into  parts  and  determine 
their  areas  alt  a2,  etc., 
the  sum  of  which  is  the 
total  area  a.  Then,  if 
vly  v2,  etc.,  be  the  mean 
velocities  in  these  areas,  FIG.  39a 

and    if    these    be    deter- 
mined by  observations,  the  discharge  is 

q  =  a1v1  -f  a2v2  +  asv3  +  etc.  (39) 


98  INSTRUMENTS  AND  OBSERVATIONS  CHAP,  iv 

Here  the  mean  velocities  may  be  roughly  found  by  observ- 
ing the  passage  of  a  surface  float  at  the  middle  of  each 
subdivision  and  multiplying  this  surface  velocity  by  0.9, 
There  are,  however,  more  precise  methods,  one  of  which 
will  be  explained  in  Art.  40,  while  others  will  be  described 
in  Chapter  X.  When  q  has  been  found  in  this  manner 
the  mean  velocity  of  the  stream  may  be  computed,  if 
desired,  by  v  =  q/a. 

Formula  (39)  applies  also  to  a  cross-section  of  any 
kind.  Thus,  let  the  pipe  of  Fig.  39b  be  divided  by  con- 
centric circles  into  the  areas  alt  a2,  a3,  a4J> 
and  let  the  mean  velocities  vlt  v2,  v3,  v4, 
be  determined  by  observation  for  each  of 
these  areas;  the  discharge  q  is  then  given 
by  (39).  Again,  in  the  conduit  of  Fig. 
118,  let  a  velocity  observation  be  taken 
at  each  of  the  97  points  marked  by  a  dot, 
these  points  being  uniformly  spaced  over  the  cross-section, 
so  that  each  of  the  areas  alf  a2,  etc.,  may  be  regarded 
as  -faa.  Then  from  (39)  the  discharge  is 

q=-faa(vl  +  v2  +  v3  +  ....  +v16)=av 

or  v  is  the  sum  of  the  individual  velocities  divided  by  97. 
In  general,  if  a  cross-section  be  divided  into  n  equal 
parts  the  mean  velocity  is  the  average  of  the  n  observed 
velocities.  This  result  is  the  more  accurate  the  greater 
the  number  of  parts  into  which  the  cross-section  is 
divided.  If  the  number  of  parts  be  infinite  and  the  water 
passing  through  each  be  called  a  filament,  the  mean  veloc- 
ity may  be  defined  as  the  average  of  the  velocities  of  all 
the  filaments. 

Prob.  39.  A  water  pipe,  3  inches  in  diameter,  is  divided  into 
three  parts  by  concentric  circles  whose  diameters  are  1,2,  and 
3  inches.  The  mean  velocities  in  these  parts  are  found  to  be 
6.2,  4.8,  and  3.0  feet  per  second.  Compute  the  discharge  and 
mean  velocity  for  the  pipe. 


ART.  40  THE  CURRENT  METER  99 


ART.  40.     THE  CURRENT  METER 

In  1790  the  German  hydraulic  engineer  Woltmann 
invented  an  apparatus  for  measuring  the  velocity  of  flow- 
ing water  which  was  later  improved  by  Darcy  and  others, 
and  is  now  extensively  used  for  streams  and  open  channels. 
This  meter  is  like  a  windmill,  having  three  or  more  vanes 
mounted  on  a  spindle,  and  so  arranged  that  the  face  of 
the  wheel  always  stands  normal  to  the  current,  the  pressure 
of  which  causes  it  to  revolve.  The  number  of  revolutions 
of  the  wheel  is  approximately  proportional  to  the  velocity 
of  the  current.  In  the  best  forms  of  instruments  the 
number  of  revolutions  made  in  a  given  time  is  determined 
by  an  apparatus  on  shore  or  in  a  boat  from  which  wires 
lead  to  the  meter  under  water;  at  every  revolution  an 
electric  connection  is  made  and  broken  which  affects  a 
dial  on  the  recording  apparatus.  The  observer  has  hence 
only  to  note  the  time  of  beginning  and  ending  of  the  ex- 
periment, and  to  read  the  number  of  revolutions  which 
have  occurred  during  the  interval.  For  a  canal  or  small 
stream  the  meter  is  best  operated  from  a  bridge;  in  large 
streams  a  boat  must  be  used. 

Fig.  40a  shows  the  recording  dial  of  a  current  meter 
which  should  be  supposed  to  be  on  a  bridge  or  in  a  boat 
with  an  electric  battery.  Fig.  406  shows  the  Price  current 
meter,  a  form  extensively  used  in  the  United  States, 
and  the  wires  connecting  the  dial  and  battery  are  seen 
to  run  down  the  standard  to  the  revolving  wheel  where 
the  electric  current  is  broken  at  each  revolution.  The 
cups  or  vanes  are  kept  facing  the  current  of  the  stream 
by  means  of  the  cross-shaped  rudder.  At  the  lower  end 
of  the  standard  is  a  heavy  lead  weight  which  serves  to 
keep  the  standard  in  a  vertical  position.  At  the  upper 
end  of  the  standard  is  seen  the  vertical  wire  which  is  held 
by  the  observer  on  the  bridge,  while  the  inclined  line 


100 


INSTRUMENTS  AND  OBSERVATIONS 


CHAP.  IV 


represents  a  cord  that  is  sometimes  used  to  give  steadiness- 
to  the  meter. 

A  current  meter  cannot  be  used  for  determining  the 
velocity  in  a  small  trough,  for  the  introduction  of  it  into 


FIG.  406 

the  cross-section  would  contract  the  area  and  cause  a 
change  in  the  velocity  in  front  of  the  wheel.  In  large 
conduits,  canals,  and  rivers  it  is,  however,  one  of  the  most 


ART.  40  THE  CURRENT  METER  101 

convenient  and  accurate  instruments.  By  holding  it  at 
a  fixed  position  below  the  surface  the  velocity  at  that 
point  is  found;  by  causing  it  to  descend  at  a  uniform 
rate  from  surface  to  bottom  the  mean  velocity  in  that  ver- 
tical is  obtained ;  and  by  passing  it  at  a  uniform  rate  over 
all  parts  of  the  cross-section  of  a  channel  the  mean  velocity 
v  is  directly  determined.  It  is  usually  attached  to  the 
end  of  a  long  chain  or  pole,  which  is  graduated  so  that 
the  depth  of  the  meter  below  the  water  surface  can  be 
directly  read.* 

To  derive  the  velocity  of  the  water  from  the  number 
of  recorded  revolutions  per  second,  the  meter  must  be 
first  rated  by  pushing  it  at  a  known  velocity  in  still  water. 
The  best  place  for  doing  this  is  in  a  navigation  canal  where 
the  water  has  no  sensible  velocity,  or  in  a  pond.  A  track 
is  built  along  the  bank  on  which  a  small  car  can  be  moved 
at  a  known  velocity,  and  on  this  car  the  observer  holds 
the  meter  in  the  water  at  the  end  of  a  pole  and  records 
the  number  of  revolutions  made  and  the  time  elapsed  in 
passing  over  a  certain  distance.  The  lowest  velocity  of 
the  car  should  be  about  0.2  feet  per  second,  and  the  highest 
about  10  feet  per  second.  It  is  always  found  that  the 
number  of  revolutions  per  minute  is  not  exactly  pro- 
portional to  the  velocity  of  the  car,  and  hence  when  the 
meter  is  placed  stationary  in  running  water  the  velocity 
of  the  water  is  not  proportional  to  the  number  of  revo- 
lutions per  second. 

From  these  observations  there  is  prepared  a  rating 
table  showing  the  velocity  of  the  water  corresponding  to 
the  number  of  revolutions  in  a  minute  or  other  given  time. 
To  make  such  a  table  the  knowrn  velocities  of  the  car  are 
taken  as  abscissas  on  cross-section  paper  and  the  numbers 
of  revolutions  as  ordinates,  and  a  point  is  plotted  corre- 
sponding to  each  observation.  A  mean  curve  may  then 

*See  U.  S.  Geological  Survey's  Water  Supply  and  Irrigation  Papers, 
No.  56  (Washington,  1901). 


102 


1  INSTRUMENTS  AND  OBSERVATIONS 


CHAP.  IV 


be  drawn  to  agree  as  closely  as  possible  with  the  plotted 
pointer,  and  from  this  curve  the  velocity  corresponding 
to  any  number  of  revolutions  can  be  taken  off.  This 
curve  may  also  be  expressed  by  an  equation  of  the  form 
V  =a  +  bn  +  cn2,  in  which  V  is  the  velocity  of  the  car  in  feet 
per  second  and  n  the  number  of  revolutions  of  the  meter 
per  minute ;  and  by  the  help  of  the  Method  of  Least  Squares 
the  constants  a,  b,  and  c  may  be  computed  (Art.  42). 

Prob.  40.   In  order  to  rate  a  certain  current  meter,  three 
observations  were  taken  in  still  water,  as  follows: 


=  2.0 


3-8 
60 


7.4  feet  per  second 
120 


Velocity  of  the  car 
Revolutions  per  minute  =  30 

Plot  these  observations  on  cross-section  paper  and  deduce, 
without  using  the  Method  of  Least  Squares,  the  relation  be- 
tween V  and  n. 


ART.  41.     THE  PITOT  TUBE 

About  1750  the  French  hydraulic  engineer  Pitot  in- 
vented a  device  for  measuring  the  velocity  in  a  stream 
by  means  of  the  velocity-head  which  it  will  produce.  In 
its  simplest  form  it  consists  of  a  bent  tube,  the  mouth  of 
which  is  placed  so  as  to  directly  face  the  current.  The 
water  then  rises  in  the  vertical  part  of  the  tube  to  a  height 


FIG.  41a 


FIG.  416 


h  above  the  surface  of  the  flowing  stream,  and  this  height 
is  theoretically  equal  to  v*/2g,  so  that  the  actual  velocity 
v  is  in  practice  approximately  equal  to  \/ 2gh.  As  con- 
structed for  use  in  streams,  Pitot 's  apparatus  consists  of 


ART.  41  THE  PITOT  TUBE  103 

two  tubes  placed  side  by  side  with  their  submerged  mouths 
at  right  angles,  so  that  when  one  is  opposed  to  the  current, 
as  seen  in  Fig.  416,  the  other  stands  normal  to  it,  and  the 
water  surface  in  the  latter  tube  hence  is  at  the  same  level 
as  that  of  the  stream.  Both  tubes  are  provided  with  cocks 
which  may -be  closed  while  the  instrument  is  immersed, 
and  it  can  be  then  lifted  from  the  water  and  the  head  h  be 
read  at  leisure.  It  is  found  that  the  actual  velocity  is 
always  less  than  */2gh,  and  that  a  coefficient  must  be  de- 
duced for  each  instrument  by  moving  it  in  still  water  at 
known  velocities.  Pitot's  tube  has  the  advantage  that 
no  time  observation  is  needed  to  determine  the  velocity, 
but  it  has  the  disadvantage  that  the  distance  h  is  usually 
very  small,  so  that  an  error  in  reading  it  has  a  large  in- 
fluence. Although  the  instrument  was  improved  by  Darcy 
in  1856  and  used  by  him  for  some  stream  measurements, 
it  was  for  a  long  time  regarded  as  having  a  low  degree  of 
precision. 

In  1888  Freeman  made  experiments  on  the  distribution 
of  velocities  in  jets  from  nozzles,  in  which  an  improved 
form  of  Pitot  tube  was  used.*  The  point  of  the  tube  facing 
the  current  was  the  tip  of  a  stylographic  pen,  the  diameter 
of  the  opening  being  about  0.006  inches.  This  point  was 
introduced  into  different  parts  of  the  jet  and  the  pressure 
caused  in  the  tube  was  measured  by  a  Bourdon  pressure 
gage  reading  to  single  pounds.  The  velocities  of  the  jets 
were  high;  for  example,  in  one  series  of  observations  on 
a  jet  from  a  ij-inch  nozzle,  the  gage  pressures  at  the  cen- 
ter and  near  the  edge  were  51.2  and  18.2  pounds  per  square 
inch,  which  correspond  to  velocity-heads  of  118.2  and  42.0 
feet,  or  to  velocities  of  87.2  and  52.0  feet  per  second.  By 
computing  the  mean  velocity  of  the  jet  from  measurements 
in  concentric  rings  (Art.  39)  and  also  from  the  measured 
discharge,  Freeman  concluded  that  any  velocity  as  deter- 
mined by  the  tube  was  smaller  than  that  computed  from 

*  Transactions  American  Society  Civil  Engineers,  1889,  vol.  21,  p.  413. 


104  INSTRUMENTS  AND  OBSERVATIONS  CHAP,  iv 

v  =  \/2gh  by  less  than  one  percent.  This  investigation 
established  the  fact  that  the  Pitot  tube  is  an  instrument 
of  great  precision  for  the  measurement  of  high  velocities. 

Experiments  on  the  flow  of  water  in  pipes,  in  which 
Pitot  tubes  were  successfully  used,  were  made  in  1897  by 
Cole  at  Terra  Haute,  and  in  1898  by  Williams,  Hubbell, 
and  Fenkell  at  Detroit.*  In  the  Detroit  experiments  the 
tube  was  introduced  into  the  pipe  through  an  opening  pro- 
vided with  a  stuffing-box,  so  that  the  point  of  the  tube 
might  be  placed  at  any  desired  position.  The  tubes  had 
openings  at  their  points  -sV-inch  in  diameter  and  other  open- 
ings of  the  same  size  on  their  sides  to  admit  the  static  pres- 
sure of  the  water.  These  latter  openings  led  to  a  common 
channel  parallel  to  that  leading  from  the  point,  and  each 
of  these  was  connected  to  a  rubber  hose  running  to  a  dif- 
ferential gage,  consisting  of  two  parallel  glass  tubes  open 
at  the  top,  where  the  difference  of  head  was  read  on  a  scale. 
In  order  to  be  able  to  deduce  the  velocities  in  the  pipe  from 
the  readings  of  the  gage,  the  Pitot  tubes  were  rated  by 
moving  them  in  still  water  at  known  velocities  as  for  the 
current  meter  (Art.  40).  Thus  a  coefficient  c  was  de- 
rived for  each  tube  for  use  in  the  formula  v  =  c\/2gh. 
This  coefficient  was  found  to  range  from  0.86  to  0.95  for 
different  tubes,  and  it  was  shown  that  it  varied  but  little 
with  the  velocity.  By  these  tubes  it  was  found  possible 
to  measure  velocities  ranging  from  i  to  6  feet  per  second 
with  a  higher  degree  of  precision  than  had  ever  before  been 
anticipated. 

Prob.  41a.  What  will  happen  if  the  point  of  a  Pitot  tube  be 
turned  down  stream? 

Prob.  416.  If  the  height  h  in  Fig.41ais  0.169  meters  and  the- 
velocity  v  is  known  to  be  1.65  meters  per  second,  show  that  the 
coefficient  of  the  tube  is  0.91. 

*  Transactions  American  Society  of  Civil  Engineers,   1902,  vol.  47,  pp. 
12,  275. 


ART.  42 


DISCUSSION  OF  OBSERVATIONS 


105 


ART.  42.     DISCUSSION  OF  OBSERVATIONS 

An  observation  is  the  recorded  result  of  a  measurement. 
All  measurements  are  affected  with  errors  due  to  imper- 
fections of  the  instrument  and  lack  of  skill  of  the  observers, 
and  the  recorded  results  contain  these  errors.  Thus,  if 
6.05,  6.02,  6.01,  and  6.04  inches  be  four  observations  on  the 
diameter  of  an  orifice,  all  of  these  cannot  be  correct  and 
probably  each  is  in  error.  The  best  that  can  be  done  is 
to  take  the  average  of  these  observations,  or  6.03  inches, 
as  the  most  probable  result,  and  to  use  this  in  the  compu- 
tations. 

An  observer  is  often  tempted  to  reject  a  measurement 
when  it  differs  from  others,  but  this  can  only  be  allowed 
when  he  is  convinced  that  a  mistake  has  been  made.  A 
mistake  is  a  large  error,  due  generally  to  carelessness,  and 
must  not  be  confounded  with  the  small  accidental  errors 
of  measurement.  When  a  series  of  observations  is  placed 
before  a  computer  he  should  never  be  permitted  to  reject 
one  of  them,  unless  there  be  some  remark  in  the  note-book 
which  casts  doubt  upon  it. 

Graphical  methods  of  discussing  and  adjusting  obser- 
vations, like  that  mentioned  in  Art.  40,  are  of  great  value 
in  hydraulic  work.  As  another  example,  the  following 
observations  made  by  Darcy  and  Bazin  on  the  flow  of  water 
in  a  rectangular  trough,  1.812  meters  wide  and  having  the 


1           \ 

^ 

3*— 

^ 

-£?** 

5                   10                 15                  20                  » 
Values  of  r 

FlG.  42a 

uniform  slope  0.049,  may  be  noted.     Water  was  allowed 
to  run  through  it  with  varying  depths,  and  for  each  depth 


106  INSTRUMENTS  AND  OBSERVATIONS  CHAP,  iv 

the  mean  velocity  (Art.  39)  and  the  hydraulic  mean  depth 
(Art.  105)  was  determined  by  measurement.  Let  v  be  the 
mean  velocity  and  r  the  hydraulic  mean  depth;  then  five 
measurements  gave  the  following  observations,  v  being  in 
meters  per  second  and  r  in  centimeters.  Let  it  be  assumed 

No.  =      i  2  3  4  5 

v=    1.73        1.98       2.17       2.33       2.46 
r  =  n.4        14.4        17.0        19.2        21.2 

that  the  relation  between  v  and  r  is  of  the  form  v  =  mrn,  and 
let  it  be  required  to  determine  the  most  probable  values  of 
m  and  n. 

For  each  of  these  observations  a  point  may  be  plotted 
on  cross-section  paper,  taking  the  values  of  v  as  ordinates 
and  those  of  r  as  abscissas,  and  a  smooth  curve  may  then 
be  drawn  so  as  to  agree  as  nearly  as  possible  with  the  points. 
Such  a  curve,  however,  is  of  little  assistance  in  determin- 
ing the  values  of*  m  and  n,  unless  the  curve  should  be  a 
straight  line  drawn  through  the  origin,  in  which  case  it  is 
plain  that  n  is  unity  and  that  m  is  the  tangent  of  the  an- 
gle that  the  line  makes  with  axis  of  abscissas.  In  this  case 
no  straight  line  can  be  drawn  approximating  to  the  points 
and  passing  through  the  origin,  but  the  plot  gives  the 
curve  shown  in  Fig.  42a.  If,  however,  the  logarithm  of 
each  side  of  the  assumed  formula  be  taken  it  becomes 

log  v  =  n  log  r  +  log  m 

which  represents  a  straight  line,  if  log  v  be  considered  as 
the  variable  ordinate  and  log  r  as  the  variable  abscissa,  log 
m  being  the  intercept  on  the  axis  of  ordinates  and  n  the 
tangent  of  the  angle  which  the  line  makes  with  the  axis 
of  abscissas.  On  plotting  the  points  corresponding  to  the 
values  of  log  v  and  log  r,  it  is  seen  that  a  straight  line  can 
be  drawn  closely  agreeing  with  the  points,  that  this  line 
cuts  the  axis  of  ordinates  at  a  distance  of  about  0.35  below 
the  origin  and  that  the  tangent  of  the  angle  made  by  it 


ART.  42 


DISCUSSION  OF  OBSERVATIONS 


107 


2 

JT** 

^~ 

^r 

^^^"^ 

^^^^ 

^-^  — 

^*^^ 

^  —  """ 

•***^ 

1                              1.5                             2 
Values  of  log.  r 

FiG.  426 

with  the  axis  of  abscissas  is  about  0.55.  Hence  ^=9.55, 
log  m=  —0.35  =  1.65,  or /;/=  0.446;  then 

log  ^=0.55  log  r  —  0.35          or         v=o.446r°'55 

is  an  empirical  formula  fc  r  computing  the  mean  velocity 
in  this  trough.  Using  the  above  values  of  r  and  computing 
those  of  v,  it  is  found  that  the  computed  and  observed 
results  agree  fairly,  the  former  being  generally  a  little 


0.40 


£0.20 


|0.20 
L. 
0,40 


smaller,  which  is  due  to  the  fact  that  only  two  significant 
figures  have  been  obtained  from  the  plot. 

There  is  a  process,  known  as  the  Method  of  Least 
Squares,  by  which  the  constants  of  an  empirical  formula 
may  be  obtained  from  observations  with  a  higher  degree 
of  precision  than  by  any  graphic  method.  Its  application 
to  the  above  case  will  here  be  given.  Let  the  simultaneous 
values  of  log  v  and  log  r  for  each  experiment  be  placed  in 
the  logarithmic  formula  as  follows: 

for  No.  i,  0.238  =  i. 05 7^  + log  m 

for  No.  2,  0.297  =  1.15 8n  +  log  m 

for  No.  3,  0.336  =  i.23on  +  log  m 

for  No.  4,  0.367  =  i. 28372  + log  m 

for  No.  5,  0.391  =i.32~6w  +  log  m 

These  five  equations  contain  two  unknown  quantities,  n 
and  log  m,  but  no  values  of  these  can  be  found  that  will 
exactly  satisfy  all  the  equations.  The  best  that  can  be 
done  is  to  find  the  values  that  have  the  greatest  degree 
of  probability,  and  these  will  satisfy  the  equations  with 


108  INSTRUMENTS  AND  OBSERVATIONS  CHAP,  iv 

the  smallest  discrepancies.  To  do  this,  let  each  equation 
be  multiplied  by  the  coefficient  of  n  in  that  equation 
and  the  results  be  added;  also  let  each  equation  be- multi- 
plied by  the  coefficient  of  log  m  in  that  equation  and  the 
results  be  added.  Thus  are  found  the  two  normal  equa- 
tions containing  the  two  unknown  quantities: 

1.998  =  7.375^  +  6.054  log  m 
1.629=6.054^+5.000  log  m 

and  the  solution  of  these  gives  ^  =  0.571  and  log  m  = 
—  0.366.  Since  —0.366  equals  1.634,  the  value  of  m  is 
0.431,  and  then 

log  v  =0.571  log  r  —  0.366     or     v=o.4^ir°-571 
is  the  empirical  formula  for  this  particular  case. 

The  Method  of  Least  Squares  is  usually  more  laborious 
than  the  graphical  method,  but  it  has  the  great  advantage 
that  its  results  are  the  most  probable  ones  that  can  be 
derived  from  the  given  data.  It  has  the  further  advantage 
that  all  computors  will  derive  the  same  results,  whereas 
in  the  graphic  method  the  results  will  usually  differ,  be- 
cause the  position  of  the  line  drawn  on  the  plot  is  affected 
by  the  different  degrees  of  judgment  and  experience  of 
the  draftsmen.  It  will  be  seen  from  Fig.  42b  that  it  is 
not  very  easy  to  determine  close  values  of  log  m  since  the 
points  are  so  far  away  from  the  origin. 

Prob.  42a.  Show  that  the  formula    v  =  o.43if°-671  reduces  to 
v=5-97*°'SIli  if  f  he  in  meters  and  v  in  meters  per  second. 

Prob.  426.   Show  that    this  formula  becomes    v  =  ^.g4rQ-5n  if 
r  be  in  feet  and  v  in  feet  per  second. 

Prob.  42c.  In  order  to  rate  a  certain  current  meter  four 
observations  were  taken  in  still  water,  as  follows: 

Velocity  of  the  car         0.7        2.4       4.7        9.3  feet  per  second 
Revolutions  of  meter     18         60         120      240  per  minute 
Find  the  values  of  a  and  b  in  the  formula  v  =  a+bn,  both  by 
plotting  and  by  the  method  of  Least  Squares. 


ART.  43 


STANDARD  ORIFICES 


109 


CHAPTER  V 
FLOW  OF  WATER  THROUGH  ORIFICES 

ART.  43.     STANDARD  ORIFICES 

Orifices  for  the  measurement  of  water  are  usually  placed 
in  the  vertical  side  of  a  vessel  or  reservoir,  but  may  also 
be  placed  in  the  base.  In  the  former  case  it  is  understood 
that  the  upper  edge  of  the  opening  is  completely  covered 
with  water;  and  generally  the  head  of  water  on  an  orifice 
is  at  least  three  or  four  times  its  vertical  height.  The 
term  "standard  orifice"  is  here  used  to  signify  that  the 
opening  is  so  arranged  that  the  water  in  flowing  from 
it  touches  only  a  line,  as  would  be  the  case  in  a  plate  of 
no  thickness.  To  secure  this  result  the  inner  edge  of 
the  opening  has  a  square  corner,  which  alone  is  touched 
by  the  water.  In  precise  experiments  the  orifice  may  be 
in  a  metallic  plate  whose  thick- 
ness is  really  small,  as  at  A  in 
the  figure,  but  more  commonly 
it  is  cut  in  a  board  or  plank, 
care  being  taken  that  the  inner 
edge  is  a  definite  corner.  It  is 
usual  to  bevel  the  outer  edges 
of  the  orifice  as  at  C,  so  that  the 
escaping  jet  may  by  no  possibility 
touch  the  edges  except  at  the  inner  corner.  The  term 
"  orifice  in  a  thin  plate  "  is  often  used  to  express  the  con- 
dition that  the  water  shall  only  touch  the  edges  of  the 
opening  along  a  line.  This  arrangement  may  be  regarded 
as  a  kind  of  standard  apparatus  for  the  measurement  of 


FIG.  43a 


HO  FLOW   THROUGH    ORIFICES  CHAP.  V 

water,  for,  as  will  be  seen  later,  the  discharge  is  modified 
if  the  inner  corner  is  rounded,  and  different  degrees  of 
rounding  give  different  discharges.  Orifices  arranged  as 
in  Fig.  43a  are  accordingly  always  used  when  water  is 
to  be  measured  by  the  use  of  orifices. 

The  contraction  of  the  jet  which  is  always  observed 
when  water  issues  from  a  standard  orifice  as  described 
above  is  a  most  interesting  and  important  phenomenon. 
It  is  due  to  the  circumstance  that  the  particles  of  water 
as  they  approach  the  orifice  move  in  converging  directions, 
and  that  these  directions  continue  to  converge  for  a  short 
distance  beyond  the  plane  of  the  orifice.  It  is  this  con- 
traction of  the  jet  that  causes  only  the  inner  corner  of 
the  orifice  to  be  touched  by  the  escaping  water.  The 
appearance  of  such  a  jet  under  steady  flow,  issuing  from 
a  circular  orifice,  is  that  of  a  clear  crystal  bar  whose 
beauty  claims  the  admiration  of  every  observer.  The 
convergence  due  to  this  cause  ceases  at  a  distance  from 
the  plane  of  the  orifice  of  about  one-half  its  diameter.. 
Beyond  this  section  the  jet  enlarges  in  size  if  it  be  directed 
upward,  but  decreases  in  size  if  it  be  directed  downward 
or  horizontally. 

The  contraction  of  the  jet  is  also  observed  in  the  case 
of  rectangular  and  triangular  orifices,  its  cross-section 

being  similar  to  that  of  the 
orifice  un"til  the  place  of  great- 
est contraction  is  passed.  Fig. 
436  shows  in  the  top  row 
(~\  <\j^p  ^{f  cross-sections  of  a  jet  from 
a  square  orifice,  in  the  middle 
row  those  from  a  triangular 

r\         s~\       /^        __      one,  and  in  the  third  row  those 

from  an  elliptical  orifice.     The 
left  hand  diagram  in  each  case 

is  the  cross-section  of  the  jet  near  the  place  of  greatest 
contraction,  while  the  following  ones  are  cross-sections 


'    ,~  n 

\j  II 


ART.  44  COEFFICIENT    OF   CONTRACTION  111 

at   greater   distances   from   the   orifice,    and  the  jets   are 
supposed  to  be  moving  horizontally,  or  nearly  so. 

Owing  to  this  contraction  the  discharge  from  a  standard 
orifice  is  always  less  than  the  theoretic  discharge.  It  is 
the  object  of  this  chapter  to  determine  how  the  theoretic 
formulas  of  Chapter  III  are  to  be  modified  so  that  they 
may  be  used  for  the  practical  purposes  of  the  measure- 
ment of  water.  This  is  to  be  done  by  the  discussion  of 
the  results  of  experiments.  It  will  be  supposed,  unless 
otherwise  stated,  that  the  size  of  the  orifice  is  small  com- 
pared with  the  cross-section  of  the  reservoir,  so  that  the 
effect  of  velocity  of  approach  may  be  neglected  (Art.  25). 

Prob.  43.  At  a  distance  from  a  circular  orifice  of  one-half 
its  diameter  a  jet  has  a  diameter  of  i  inch  and  a  velocity  of 
1 6  feet  per  second.  If  it  be  directed  vertically  downward, 
what  is  the  diameter  of  a  section  4  feet  lower?  If  it  be  directed 
vertically  upward,  what  is  the  diameter  of  a  section  3  feet 
higher  ? 

ART.  44.     COEFFICIENT  OF  CONTRACTION 

The  coefficient  of  contraction  is  the  number  by  which 
the  area  of  the  orifice  is  to  be  multiplied  in  order  to  give 
the  area  of  the  section  of  the  jet  at  a  distance  from  the 
plane  of  the  orifice  of  about  one-half  its  diameter.  Thus, 
if  c'  be  the  coefficient  of  contraction,  a  the  area  of  the 
orifice,  and  a'  the  area  of  the  contracted  section  of  jet, 

a'=c'a  (44) 

The   coefficient   of   contraction   for   a   standard   orifice   is 
evidently  always  less  than  unity. 

The  only  direct  method  of  finding  the  value  of  c'  is 
to  measure  by  calipers  the  dimensions  of  the  least  cross- 
section  of  the  jet.  The  size  of  the  orifice  can  usually  be 
determined  with  precision,  and  with  care  almost  an  equal 
precision  in  measuring  the  jet.  To  find  c'  for  a  circular 


112  FLOW   THROUGH    ORIFICES  CHAP.  V 

orifice  let  d  and  df  be  the  diameters  of  the  section  a  and 
a';  then 


Therefore  the  coefficient  of  contraction  is  the  square  of 
the  ratio  of  the  diameter  of  the  jet  to  that  of  the  orifice. 
The  first  measurements  were  made  by  Newton*  who 
found  the  ratio  of  d'  to  d  to  be  21/25  which  gives  for  c 
the  value  0.73.  The  experiments  of  Bossut  gave  from 
0.66  to  0.67;  a,nd  Michelotti  found  from  0.57  to  0.624 
with  a  mean  of  0.6  1.  Eytelwein  gave  0.64  as  a  mean 
value,  and  Weisbach  mentions  0.63. 

The  following  mean  value  will  be  used  in  this  book 
and  it  should  be  kept  in  mind  by  the  student: 

Coefficient  of  contraction  c'  =0.62 

or,  in  other  words,  the  minimum  cross-section  of  the  jet 
is  62  percent  of  that  of  the  orifice.  This  value,  however, 
undoubtedly  varies  for  different  forms  of  orifices  and  for 
the  same  orifice  under  different  heads,  but  little  is  known 
regarding  the  extent  of  these  variations  or  the  laws  that 
govern  them.  Probably  c'  is  slightly  smaller  for  circles 
'than  for  squares,  and  smaller  for  squares  than  for  rect- 
angles; particularly  if  the  rectangle  be  long  compared  with 
its  width.  Probably  also  c'  is  larger  for  low  heads  than 
for  high  heads. 

Prob.  44.  The  diameter  of  a  circular  orifice  is  1.995  inches. 
Three  measurements  of  the  diameter  of  the  contracted  section 
of  the  jet  gave  1.55,  1.56,  and  1.59  inches.  Find  the  mean 
coefficient  of  contraction. 

ART.  45.     COEFFICIENT  OF  VELOCITY 

The  coefficient  of  velocity  is  the  number  by  which  the 
theoretic  velocity  of  flow  from  the  orifice  is  to  be  multiplied 
in  order  to  give  the  actual  velocity  at  the  least  cross-section 

*  Philosophise  naturalis   principia    mathematica,  1687,  Book  II,  prop- 
osition 36. 


ART.  45  COEFFICIENT    OF   VELOCITY  113 

of  the  jet.  Thus,  if  <7t  be  the  coefficient  of  velocity,  V  the 
theoretic  velocity  due  to  the  head  on  the  center  of  the  ori- 
fice, and  v  the  actual  velocity  at  the  contracted  section, 


(45) 

The  coefficient  of  velocity  must  be  less  than  unity,  since  the 
force  of  gravity  cannot  generate  a  greater  velocity  than  that 
due  to  the  head. 

The  velocity  of  flow  at  the  contracted  section  of  the  jet 
cannot  be  directly  measured.  To  obtain  the  value  of  the 
coefficient  of  velocity,  indirect  observations  have  been 
taken  on  the  path  of  the"  jet.  Referring  to  Art.  27,  it  will 
be  seen  that  when  a  jet  flows  from  an  orifice  in  the  vertical 
side  of  a  vessel  it  takes  a  path  whose  equation  is  y  =gx*/2V2, 
in  which  x  and  y  are  the  co-ordinates  of  any  point  of  the 
path  measured  from  vertical  and  horizontal  axes,  and  v 
is  the  velocity  at  the  origin.  Now  placing  for  v  its  value 
c1\/2gh,  and  solving  for  clt  gives 

c±  =x/2\/hy 

Therefore  c±  becomes  known  by  the  measurement  of  the 
head  h  and  the  co-ordinates  x  and  y.  In  making  this  experi- 
ment it  would  be  well  to  have  a  ring,  a  little  larger  than  the 
jet,  supported  by  a  stiff  frame  which  can  be  moved  until 
the  jet  passes  through  the  ring.  The  flow  of  water  can 
then  be  stopped,  and  the  co-ordinates  of  the  center  of  the 
ring  determined.  By  placing  the  ring  at  different  points 
of  the  path  different  sets  of  co-ordinates  can  be  obtained. 
The  value  of  x  should  be  measured  from  the  contracted 
section  rather  than  from  the  orifice,  since  v  is  the  velocity 
at  the  former  point  and  not  at  the  latter. 

By  this  method  of  the  jet  Bossut  in  two  experiments 
found  for  the  coefficient  of  velocity  the  values  0.974  and 
0.980,  Michelotti  in  three  experiments  obtained  0.993, 
0.998,  and  0.983,  and  Weisbach  deduced  0.978.  Great 
precision  cannot  be  obtained  in  these  determinations,  nor 


114  FLOW   THROUGH    ORIFICES  CHAP.  V 

indeed  is  it  necessary  for  the  purposes  of  hydraulic  investi- 
gation that  c1  should  be  accurately  known  for  standard 
orifices.  As  a  mean  value  the  following  may  be  kept  in  the 
memory : 

Coefficient  of  velocity  ct  =0.98 

or,  the  actual  velocity  of  flow  at  the  contracted  section  is  98 
percent  of  the  theoretic  velocity.  The  value  of  cl  for  the 
standard  orifice  is  greater  for  high  than  for  low  heads,  and 
may  probably  often  exceed  0.99. 

Another  method  of  finding  the  coefficient  cl  is  to  place 
the  orifice  horizontal  so  that  the  jet  will  be  directed  ver- 
tically upward,  as  in  Fig.  22.  The  height  to  which  it  rises 
is  the  velocity-head  h0=v2/2g,  in  which  v  is  the  actual 
velocity  cl\/2gh.  Accordingly,  h0  =  clh,  from  which  c^ 
may  be  computed.  For  example,  if,  under  a  head  of  23 
feet,  a  jet  rises  to  a  height  of  22  feet,  the  coefficient  of  ve- 
locity is 

cl  =  \/h()/h  =  \/22/23  =0.978 

This  method,  however,  fails  to  give  good  results  for  high 
velocities,  owing  to  the  resistance  of  the  air,  and  more- 
over it  is  impossible  to  measure  with  precision  the  height  hQ. 

For  a  vertical  orifice  Poncelet  and  Lesbros  found,  in 
1828,  that  the  coefficient  c±  was  sometimes  slightly  greater 
than  unity,  and  this  was  confirmed  by  Bazin  in  1893.  This 
is  probably  due  to  the  fact  that  the  head  is  greater  for  the 
lower  part  of  the  orifice  than  for  the  upper  part  and  hence 
\/2gh  does  not  represent  the  true  theoretic  velocity.  The 
same  experimenters  found  no  instance  of  a  horizontal  ori- 
fice where  the  coefficient  exceeded  unity. 

Prob.  45a.  To  what  height  will  a  jet  rise  when  ^  =  0.9  and 
v  =  4  feet  per  second? 

Prob.  456.  The  range  of  a  jet  is  13.5  feet  on  a  horizontal 
plane  2.82  feet  below  the  orifice  which  is  under  a  head  of  14.38 
feet.  Compute  the  coefficient  of  velocity. 


ART.  46  COEFFICIENT    OF    DISCHARGE  115 


ART.  46.     COEFFICIENT  OF  DISCHARGE 

The  coefficient  of  discharge  is  the  number  by  which  the 
theoretic  discharge  is  to  be  multiplied  in  order  to  obtain  the 
actual  discharge.  Thus,  if  c  be  the  coefficient  of  discharge, 
Q  the  theoretical  and  q  the  actual  discharge  per  second, 

q-cQ  (46), 

Here  also  the  coefficient  c  is  a  number  less  than  unity. 

The  coefficient  of  discharge  can  be  accurately  found  by 
allowing  the  flow  from  an  orifice  to  fall  into  a  vessel  of  con- 
stant cross-section  and  measuring  the  heights  of  water  by 
the  hook  gage  (Art.  35).  Thus  q  is  known,  and  Q  having 
been  computed, 

(46), 


For  example,  a  circular  orifice  of  o.i  feet  diameter  was  kept 
under  a  constant  head  of  4.677  feet;  during  5  minutes  and 
32^  seconds  the  jet  flowed  into  a  measuring  vessel  which 
was  found  to  contain  27.28  cubic  feet.  Here  the  actual 
discharge  was 

2  =  27.28/332.2  =0.08212  cubic  feet  per  second. 
The  theoretic  discharge,  from  formula  (23)  is 


Q  =  7rXo.o52X8.o2V/4.677  =0.1361  cubic  feet  per  second. 
Then  the  coefficient  of  discharge  is  found  to  be 
c  =  o.  08212/0. 1361  =0.604 

In  this  manner  thousands  of  experiments  have  been  made 
upon  different  forms  of  orifices  under  different  heads,  for 
accurate  knowledge  regarding  this  coefficient  is  of  great  im- 
portance in  practical  hydraulic  work. 

The  following  articles  contain  values  of  the  coefficient  of 
discharge  for  different  kinds  of  orifices,  and  it  will  be  seen 


116  FLOW   THROUGH    ORIFICES  CHAP.  V 

that  in  general  c  is  greater  for  low  heads  than  for  high  heads r 
greater  for  rectangles  than  for  squares,  and  greater  for 
squares  than  for  circles.  Its  value  ranges  from  0.59  to  0.63 
or  higher,  and  as  a  mean  to  be  kept  in  mind  the  following 
value  may  be  stated : 

Coefficient  of  discharge  c  =  o.6i 

or,  the  actual  discharge  from  a  standard  orifice  is,  on  the- 
average,  about  61  percent  of  the  theoretic  discharge. 

The  coefficient  c  may  be  expressed  in  terms  of  the  coef- 
ficients c'  and  c±.  Let  a  and  a'  be  the  areas  of  the  orifice  and 
the  cross-section  of  the  contracted  jet,  and  Q  and  q  the  theo- 
retic and  actual  discharge  per  second.  Then,  since  a' /a  =  c',. 


q      a 

C  =  ~z.  = .  *  i 

Q       aV2gh       ^   J 

and  therefore  the  coefficient  of  discharge  is  the  product  of 
the  coefficients  of  contraction  and  velocity. 

Prob.  46.  The  diameter  of  a  contracted  circular  jet  was. 
found  to  be  0.79  inches,  the  diameter  of  the  orifice  being  i  inch.. 
Under  a  head  of  4  feet  the  actual  discharge  per  minute  was. 
found  to  be  3.21  cubic  feet.  Find  the  coefficient  of  velocity. 


ART.  47.     CIRCULAR  VERTICAL  ORIFICES 

Let  a  circular  orifice  of  diameter  d  be  in  the  side  of  a, 
vessel  and  let  h  be  the  head  of  water  on  its  center.  Then,, 
from  Art.  22,  the  theoretic  mean  velocity  is  \/2gh,  and  from 
Art.  23  the  theoretic  discharge  is 


which  applies  when  h  is  large  compared  with  d. 

To  deduce  a  more  exact  formula,  let  the  radius  of  the 


ART.  47  CIRCULAR    VERTICAL    ORIFICES  117 

circle  be  r,  and  let  an  elementary  strip  be  drawn  at  a  dis- 

tance y  above  the  center  ;  the  length  . 

of    this    is    2VV2  —  j2,    its    area    is  1 

2dyVr2  —  y2,  and  the  head  upon  it 

is    h  —  y.     Then    the    theoretic    dis- 

charge through  this  strip  is 

dQ  =  2  dyVr2-y*\/2g(h-y) 

To  integrate   this  (h-y)l  is  to  be 

expanded  by  the  binomial  formula.     Then  it  may  be  written 


Each  term  of  this  expression  is  now  integrable,  and  taking 
the  limits  of  y  as  +  r  and  —  r  the  entire  circle  is  covered,  and 
Q  is  found.  Finally,  replacing  r  by  \d  there  results 


which  is  the  theoretic  discharge  from  the  circular  orifice. 

It  is  plain  that  this  formula  gives  values  which  are 
always  less  than  those  found  from  the  approximate  formula 
of  the  first  paragraph.  Thus  for  h=d  the  quantity  in  the 
parenthesis  is  0.992  and  for  h  =  2d  it  is  0.998.  Hence  the 
error  in  using  the  approximate  formula  is  less  than  three- 
tenths  of  one  percent  when  the  head  on  the  center  of  the 
orifice  is  greater  than  twice  its  diameter. 

For  most  cases,  then,  the  actual  discharge  from  a  cir- 
cular vertical  orifice  of  area  a  may  be  computed  from 

q=c.a\/2gh  =  8.o2ca\/h  (47)t 

in  which  c  is  the  coefficient  of  discharge.  When  h  is  smaller 
than  two  or  three  times  the  diameter  of  the  orifice,  and 
when  precision  is  required, 

q  =  1  1  —  0.078127-^  —  0.0003067-;)  &.o2ca\/h        (47)3 


118  FLOW   THROUGH    ORIFICES  CHAP.  V 

is  the  formula  to  be  used.  Here  a  may  be  taken  from  Table 
51  for  the  given  diameter  expressed  in  feet,  h  is  to  be  taken. 
in  feet,  and  then  q  will  be  in  cubic  feet  per  second. 

Table  17  gives  values  of  c  for  circular  orifices  as  deter- 
mined by  Hamilton  Smith  in  a  discussion  of  all  the  best 
experiments.*  They  apply  only  to  standard  orifices  with 
definite  inner  edges.  The  table  shows  that  the  coefficient 
of  discharge  decreases  as  the  size  of  the  orifice  increases, 
and  that  in  general  it  rlso  decreases  as  the  head  increases. 
In  this  table  the  coefficients  found  above  the  horizontal  lines 
in  the  last  three  columns  are  to  be  used  in  the  exact  formula 
(47)  2  and  all  others  in  the  approximate  formula  (47)  v 

For  example,  let  it  be  required  to  find  the  discharge 
through  a  standard  circular  orifice,  2  inches  in  diameter, 
under  a  head  of  2.35  feet.  First,  2  inches  =  0.1667  feet,  and 
by  interpolation  in  Table  17  the  coefficient  c  is  found  to 
be  0.602.  Next,  from  Table  51  the  area  a  is  0.02182  square 
feet.  Then  formula  (47)!  gives  the  discharge  q  as  0.161 
cubic  feet  per  second.  As  the  coefficient  is  probably  liable 
to  an  error  of  one  or  two  units  in  the  last  figure,  the  third 
figure  of  this  value  of  q  is  subject  to  the  same  uncertainty. 

Prob.  47a.  Find  from  the  table  the  coefficient  of  discharge 
for  an  orifice,  2  inches  in  diameter,  under  a  head  of  1.75  feet. 

Prob.  476.  Compute  the  probable  actual  discharge  from  an 
orifice,  8  inches  in  diameter,  under  a  head  of  15  inches. 

ART.  48.     SQUARE  VERTICAL  ORIFICES 

If  the  size  of  an  orifice  in  the  side  of  a  vessel  be  small 
com-  •  1  with  the  head,  the  theoretic  velocity  of  the 
outiVi  =  water  may  be  taken  as  V2gh9  where  h  is  the 
heai  <  iie  center  of  the  orifice.  For  a  rectangular 

orifk  e  under  this  condition  the  theoretic  discharge  is 


*  Hydraulics  (London  and  New  York,  1886),  page  59. 


ART.  48  SQUARE   VERTICAL    ORIFICES  119 

where  b  is  the  width  and  d  the  depth  of  the  orifice.     When 
b  is  equal  to  d  the  rectangle  becomes  a  square. 

To  deduce  a  more  exact  formula,  let  k^  be  the  head 
on  the  upper  edge  of  the  orifice  and  k2  that  on  the  lower 
edge.  Consider  an  elementary  strip  . 
of  area  b  .  dy  at  a  depth  y  below  the 
water  level.  The  .velocity  of  flow 
through  this  elementary  strip  is  \/2gy 
and  the  theoretic  discharge  per  second 
through  it  is 


Integrating  this  between  the  limits  h2  and  k^  there  results 


which  is  the  true  theoretic  discharge  from  the  orifice. 

To  ascertain  the  error  caused  by  using  the  approximate 
formula,  let  k  be  the  head  on  the  center  of  the  rectangle; 
then  h2  =  h  +  $d  and  hl=h  —  %d.  Developing  by  the  bino- 
mial formula  the  values  of  k$  and  h^t  the  last  formula 
Incomes 


and  this  shows  that  the  discharge  computed  by  using 
the  approximate  formula  is  always  too  great.  For  h=d, 
the  quantity  in  the  parenthesis  is  0.989,  and  for  h  =  2d 
it  is  0.997.  Accordingly,  the  error  of  the  approximate 
formula  is  only  three-tenths  of  one  percent  when  the 
head  on  the  center  of  the  rectangle  is  twice  the  depth 
of  the  orifice. 

For   most   cases,    then,    the   actual   discharge   from   a 
square  vertical  orifice  may  be  computed  from 

q=c.  b*Vl2gh  =  S.o2cb2V~h 


120  FLOW  THROUGH  ORIFICES  CHAP,  v 

where  b  is  the  side  of  the  square  and  c  is  the  coefficient 
of  discharge.  When  h  is  smaller  than  two  or  three  times 
the  side  of  the  orifice,  and  when  precision  is  required, 

-fci*)  (48), 


is  the  formula  to  be  used.  The  linear  quantities  are 
to  be  taken  in  feet,  and  then  q  will  be  in  cubic  feet  per 
second. 

Table  19  gives  values  of  the  coefficient  c  for  standard 
square  orifices,  taken  from  a  more  extended  one  formed 
by  Hamilton  Smith  in  1886  by  the  discussion  of  all  the 
best  experiments.  It  is  seen  that  the  coefficient  decreases 
as  the  size  of  the  orifice  increases  and  as  the  head  increases. 
Comparing  this  table  with  Table  17  it  is  seen  that  the 
coefficient  of  discharge  for  a  square  is  always  slightly 
larger  than  that  for  a  circle  having  a  diameter  equal  to 
the  side  of  the  square.  The  values  above  the  horizontal 
lines  in  the  last  three  columns  are  to  be  used  in  the  exact 
formula  (48)2  when  precision  is  required,  and  all  other  values 
in  the  approximate  formula  (48)  A. 

There  are  few  recorded  experiments  on  large  square 
orifices:  Ellis  measured  the  discharge  from  a  vertical 
orifice  2  feet  square*  and  deduced  the  following  coeffi- 
cients for  use  in  the  ~  approximate  formula: 

for  /i  =  2.  07  feet,  c  =  o.6n 
for  /&  =  3.05  feet,  £  =  0.597 
for  /*  =  3.  54  feet,  c  =0.604 

which  indicate  that  a  mean  value  of  0.60  may  be  used 
for  large  square  orifices  under  low  heads. 

Prob.  48a.  Find  from  the  table  the  coefficient  for  an  orifice 
3  inches  square  when  the  head  on  its  center  is  1.8  feet. 

Prob.  486.  Compute  the  probable  actual  discharge  from  a 
vertical  orifice  one  foot  square  when  the  head  on  its  upper  edge 
is  one  foot. 

*  Transactions  American  Society  Civil  Engineers,  1876,  vol.  5,  p.  92. 


ART.  49  RECTANGULAR   VERTICAL    ORIFICES  121 

ART.  49.     RECTANGULAR  VERTICAL  ORIFICES 

The  theoretic  formulas  of  Art.  48  apply  to  rectangles 
of  width  b  and  depth  d,  and  the  approximate  formula 
for  computing  the  actual  discharge  is 

q  =  cbd\/2gh  =  8.02cbdVh  (49) 

in  which  c  is  the  coefficient  of  discharge,  b  the  width  and 
d  the  depth  of  the  rectangular  orifice,  and  h  the  head  on 
its  center. 

Table  21  gives  values  of  the  coefficient  c  which  have 
been  compiled  and  rearranged  from  the  discussion  given 
by  Fanning.*  It  is  seen  that  the  variation  of  c  with 
the  head  follows  the  same  law  as  for  circles  and  squares. 
It  is  also  seen  that  for  a  rectangle  of  constant  breadth 
the  coefficient  increases  as  the  depth  decreases,  from 
which  it  is  to  be  inferred  that  for  a  rectangle  of  constant 
depth  the  coefficient  increases  with  the  breadth,  and 
this  is  confirmed  by  other  experiments.  The  value  of  c 
for  a  rectangular  orifice  is  seen  to  be  only  slightly  larger 
than  that  for  a  square  whose  side  is  equal  to  the  depth 
of  the  rectangle.  All  the  coefficients  in  this  table  are  for 
the  above  approximate  formula,  since  that  formula  was 
used  in  computing  them. 

A  comparison  of  the  values  of  c  for  the  orifice  one  foot 
square  with  those  in  the  last  article  shows  that  the  two 
sets  of  coefficients  disagree,  these  being  about  one  percent 
greater.  This  is  probably  due  to  the  less  precise  character 
and  smaller  number  of  experiments  from  which  they 
were  deduced. 

Prob.  49a.  What  constant  head  is  required  to  discharge 
5  cubic  feet  of  water  per  second  through  an  orifice  3  inches 
deep  and  12  inches  long? 

Prob.  496.  What  is  a  probable  coefficient  of  discharge  for 
an  orifice  3  inches  deep  and  6  inches  long,  the  head  on  the 
upper  edge  being  6  inches? 

*  Treatise  on  Water  Supply  Engineering  (New  York,  1888),  p.  205. 


122  FLOW  THROUGH  ORIFICES  CHAP,  v 


ART.  50.     THE  MINER'S  INCH 

The  miner's  inch  may  be  roughly  defined  to  be  the 
quantity  of  water  which  will  flow  from  a  vertical  standard 
orifice  one  inch  square,  when  the  head  on  the  center  of 
the  orifice  is  6J  inches.  From  Table  19  the  coefficient  of 
discharge  is  seen  to  be  about  0.623,  and  accordingly  the 
actual  discharge  from  the_orifice  in  cubic  feet  per  se'cond 
is  g=Ti*Xo.623  X8.o2\/6. 5/12  =0.0255  and  the  discharge 
in  one  minute  is  60X0.0255  =  1.53  cubic  feet.  The  mean 
value  of  one  miner's  inch  is  therefore  about  1.5  cubic 
feet  per  minute. 

The  actual  value  of  the  miner's  inch,  however,  differs 
considerably  in  different  localities.  Bowie  states  that  in 
different  counties  of  California  it  ranges  from  1.20  to 
1.76  cubic  feet  per  minute.*  The  reason  for  these  varia- 
tions is  due  to  the  fact  that  when  water  is  bought  for 
mining  or  irrigating  purposes  a  much  larger  quantity 
than  one  miner's  inch  is  required,  and  hence  larger  orifices 
than  one  square  inch  are  needed.  Thus  at  Smarts ville 
a  vertical  orifice  or  module  4  inches  deep  and  250  inches 
long,  with  a  head  of  7  inches  above  the  top  edge,  is  said 
to  furnish  1000  miner's  inches.  Again,  at  Columbia 
Hill,  a  module  12  inches  deep  and  12!  inches  wide,  with 
a  head  of  6  inches  above  the  upper  edge,  is  said  to  furnish 
200  miner's  inches.  In  Montana  the  customary  method 
of  measurement  is  through  a  vertical  rectangle,  one  inch 
deep,  with  a  head  on  the  center  of  the  orifice  of  4  inches, 
and  the  number  of  miner's  inches  is  said  to  be  the  same 
as  the  number  of  linear  inches  in  the  rectangle;  thus 
under  the  given  head  an  orifice  one  inch  deep  and  60  inches 
long  would  furnish  60  miner's  inches.  The  discharge 
of  this  is  said  to  be  about  1.25  cubic  feet  per  minute,  or 
75  cubic  feet  per  hour.  , 

*  Treatise  on  Hydraulic  Mining  (New  York,  1885),  p.  268. 


ART.  50  THE  MINER'S  INCH  123 

The  following  are  the  values  of  the  miner's  inch  in 
different  parts  of  the  United  States:  In  California  and 
Montana  it  is  established  by  law  that  40  miner's  inches 
shall  be  the  equivalent  of  one  cubic  foot  per  second,  and 
in  Colorado  38.4  miner's  inches  is  the  equivalent.  In 
other  States  and  Territories  there  is  no  legal  value,  but 
by  common  agreement  50  miner's  inches  is  the  equivalent 
of  one  cubic  foot  per  second  in  Arizona,  Idaho,  Nevada, 
and  Utah;  this  makes  the  miner's  inch  equal  to  1.2  cubic 
feet  per  minute. 

A  module  is  an  orifice  which  is  used  in  selling  water, 
and  which  under  a  constant  head  is  to  furnish  a  given 
number  of  miner's  inches,  or  a  given  quantity  per  second. 
The  size  and  proportions  of  modules  vary  greatly  in 
different  localities,  but  in  all  cases  the  important  feature 
to  be  observed  is  that  the  head  should  be  maintained 
nearly  constant  in  order  that  the  consumer  may  receive 
the  amount  of  water  for  which  he  bargains  and  no  more. 

The  simplest  method  of  maintaining  a  constant  head 
is  by  placing  the  module  in  a  chamber  which  is  provided 
with  a  gate  that  regulates  the  entrance  of  water  from 
the  main  reservoir  or  canal.  This  gate  is  raised  or  lowered 
by  an  inspector  once  or  twice  a  day  so  as  to  keep  the 
surface  of  the  water  in  the  chamber  at  a  given  mark. 
This  plan  is  a  costly  one,  on  account  of  the  wages  of  the 
inspector,  except  in  works  where  many  modules  are  used 
and  where  a  daily  inspection  is  necessary  in  any  event, 
and  it  is  not  well  adapted  to  cases  where  there  are  frequent 
and  considerable  fluctuations  in  the  surface  of  the  water 
in  the  feeding  canal. 

Numerous  methods  have  been  devised  to  secure  a 
constant  head  by  automatic  appliances;  for  instance, 
the  gate  which  admits  water  into  the  chamber  may  be 
made  to  rise  and  fall  by  means  of  a  float  upon  the  surface ; 
the  module  itself  may  be  made  to  decrease  in  size  when 


124  FLOW  THROUGH  ORIFICES  CHAP,  v 

the  water  rises,  and  to  increase  when  it  falls,  by  a  gate 
or  by  a  tapering  plug  which  moves  in  and  out  and  whose 
motion  is  controlled  by  a  float.  These  self-acting  con- 
trivances, however,  are  liable  to  get  out  of  order,  and 
require  to  be  inspected  more  or  less  frequently.  Another 
method  is  to  have  the  water  flow  over  the  crest  of  a  weir 
as  soon  as  it  reaches  a  certain  height.* 

The  use  of  the  miner's  inch,  or  of  a  module,  as  a  standard 
for  selling  water,  is  awkward  and  confusing,  and  for  the 
sake  of  uniformity  it  is  greatly  to  be  desired  that  water 
should  always  be  bought  and  sold  by  the  cubic  foot  per 
second.  Only  in  this  way  can  comparisons  readily  be 
made,  and  the  consumer  be  sure  of  obtaining  exact  value 
for  his  money. 

Prob.  50.  If  a  miner's  inch  be  1.57  cubic  feet  per  minute, 
how  many  miner's  inches  will  be  furnished  by  a  module  2 
inches  deep  and  50  inches  long  with  a  head  of  6  inches  above 
the  upper  edge? 

ART.  51.     VELOCITY  OF  APPROACH 

It  was  shown  in  Art.  25  that  the  theoretic  velocity 
of  flow  from  an  orifice  is  greater  than  \/2gh  when  the  ratio 
of  the  cross-section  of  the  orifice  to  that  of  the  vessel  or 
tank  is  not  small.  The  same  is  true  for  the  actual  velocity, 
but  formula  (25)  j  must  be  modified  because  it  takes  no 
account  of  the  contraction  of  the  jet.  Let  v  be  the  ve- 
locity at  the  contracted  section  of  the  jet  and  a'  the  area 
of  that  section  ;  let  v±  be  the  velocity  through  the  horizontal 
cross-section  A  of  the  vessel;  then  a'v  =  Avl.  But  if  a 
be  the  area  of  the  orifice  and  c'  the  coefficient  of  contraction, 
then  a'  equals  ac'  and  hence  c'av^Av^  Now  the  effective 
head  on  the  orifice  is 


*Foote,  Transactions  American  Society  of  Civil  Engineers,   1887,  voL 
i6,p.  134. 


ART.  51  VELOCITY  OF  APPROACH  125 


and  the  velocity  v  is  given  by  cl\/2gH  where  cl  is  the 
coefficient  of  velocity.  Substituting  in  the  last  equation 
v2/2gc^  for  H  and'  c'va/A  for  vlt  and  noting  that  c^c'  is 
equal  to  the  coefficient  of  discharge  c,  it  reduces  to 


which  is  the  velocity  of  the  jet  at  a  section  distant  from 
the  orifice  about  one-half  its  diameter.  The  discharge  q 
is  found  by  multiplying  this  by  the  area  c'a  of  that  cross- 
section,  whence 


is  the  formula  for  the  actual  discharge,  and  this  includes 
no  coefficient  except  that  of  discharge. 

These  formulas  apply  to  orifices  of  any  kind,  and 
when  c  equals  unity  they  reduce  to  the  theoretic  expressions 
established  in  Art.  25.  When  a/  A  is  less  than  1/5,  as 
is  almost  always  the  case  in  practice,  the  last  formula 
may  be  written,  with  sufficient  precision, 

(51), 


For  example,  let  a  square  tank,  4X4  feet  in  horizontal 
cross-section,  have  a  standard  square  orifice  one  square 
foot  in  area,  and  let  the  head  on  its  center  be  16  feet. 
From  Table  19  the  coefficient  of  discharge  is  0.60,  and  the 
formula  gives 

q  =  (i  +0.0007)  Xo.6oX  i  X8.02  X4 

=  19.3  cubic  feet  per  second. 

For  this  case  it  is  seen  that  the  influence  of  velocity  of 
approach  is  expressed  by  the  addition  of  0.0007  to  unity, 
which  is  an  increase  of  less  than  one-tenth  of  one  percent. 
In  general  the  increase  in  discharge  due  to  velocity  of 
approach  is  expressed,  if  a/  A  be  not  greater  than  1/5, 
by  &3a(a/A)2\/~2gh. 


126  FLOW  THROUGH  ORIFICES  CHAP,  v 

A  common  case  is  that  where  the  vessel  or  tank  is 
of  large  horizontal  and  small  vertical  cross-section,  and 
where  '  the  water  approaches  the  oriffce  with  a  horizontal 
velocity,  as  in  a  canal  or  conduit.  Here  let  A  be  the  area 
of  the  vertical  cross-section  of  the  vessel,  a  the  area  of  the 
orifice  and  h  the  head  on  its  center.  Then,  if  the  head  h 
be  large  compared  with  the  depth  of  the  orifice,  the  same 
reasoning  applies  as  in  Art.  25,  the  theoretic  velocity  is 
given  by  (25)  t  and  the  actual  discharge  by  (51)2. 

When  the  head  h  is  not  large  let  ht  and  ht  be  the  heads 
on  the  upper  and  lower  edges  of  the  orifice,  which  is 

taken  as  rectangular  and 
of  the  width  b.  Let  v  be 
the  velocity  of  approach, 
which  is  regarded  as  uni- 
form over  the  area  A. 
Then,  by  the  same  .  reasoii- 
FIG.  51  ing  as  that  in  Art.  25,  the 

theoretic  velocity  in  the  plane  of  the  orifice  at  the  depth 
y  below  the  water  level  is  given  by  V2  =  2gy  +  v2.  The 
theoretic  discharge  through  an  elementary  strip  of  the 
length  b  and  the  depth  dy  now  is 


and,  by  integration  between  the  limits  h2  and  hlt  the  total 
theoretic  discharge  is  found.  If  v2/2g  be  replaced  by 
hot  the  head  which  would  cause  the  velocity  v,  the  theoretic 
discharge  is 

Q  =  |6V^[(/z2  +  ^0)l-(^  +  ^o)1]  (51), 

and  the  actual  discharge  q  is  found  by  multiplying  this 
by  a  coefficient  of  discharge.  When  there  is  no  velocity 
of  approach  the  formula  reduces  to  that  found  in  Art. 
49  for  this  case. 

Prob.  51a.  If  n  be  a  small  quantity  compared  with  unity, 
show  that  (i  +  w)*  =  i  +  Jn,  and  that  i/(i-f-n)  =  i  —  n.  Deduce 
formula  (51  )3  from  (51  )2. 


ART.  52  SUBMERGED    ORIFICES  127 

Prob.  516.  In  the  case  of  horizontal  approach,  as  seen  in 
Fig.  51,  let  6  =  4  feet,  /*2  =  o.8  feet,  hl  =  o1  ^  =  2.5  feet  per  second, 
and  c  =  o.6.  Show  that  the  discharge  is  10.5  cubic  feet  per 
second. 


ART.  52.     SUBMERGED  ORIFICES 

It  is  shown  in  Art.  24  that  the  effective  head  h  which 
causes  the  flow  from  a  submerged  orifice  is  the  difference 
in  level  between  the  two  water  surfaces.  The  discharge 
from  such  an  orifice,  its  inner  edge  being  a  sharp  definite 
one  as  in  Fig.  43a,  has  been  found  by  experiment  to  be 
slightly  less  than  when  the  flow  oc- 
curs freely  into  the  air,  and  hence 
the  values  of  the  coefficients  of  dis- 
charge are  slightly  smaller  than  those 
given  in  Tables  17-21.  For  large 
orifices  and  large  heads  the  difference  - 

is  very  small,  and  for  orifices  one  inch 

.     1  ,    .  FIG.  52 

square    under    six    inches    head    it    is 

about  2  percent.     In  all  cases  of  submerged  orifices  the 
discharge  is  to  be  found  from  q  =  ca\/2gh. 

Table  22  gives  values  of  the  coefficient  of  discharge 
for  submerged  orifices  as  determined  from  experiments 
made  by  Hamilton  Smith  in  1884.  The  depth  of  sub- 
mergence of  the  orifices  varied  from  0.57  to  0.73  feet. 
As  a  mean  value  of  the  coefficient  of  discharge  for  standard 
submerged  orifices  0.6  is  frequently  used. 

The  theoretic  discharge  from  a  submerged  orifice  is 
the  same  for  the  same  effective  head  h  whatever  be  its 
distance  below  the  lower  water  level.  The  theoretic 
velocity  in  all  parts  of  the  orifice  is  the  same,  as  may  be 
proved  from  Fig.  52,  where  the  triangles  A  CD  and  BCE 
represent  the  distribution  of  pressure  on  AC  and  BC 
when  the  orifice  is  closed  (Art.  17).  Making  CF  equal  to 
CE  and  drawing  BF  the  unit-pressure  on  BC  is  seen  to 


128  FLOW  THROUGH  ORIFICES  CHAP,  v 

have  the  constant  value  DF.  Now  when  the  orifice  is 
opened  the  velocity  at  any  point  depends  on  the  unit- 
pressure  there  acting,  as  seen  by  (24)  lf  and  accordingly 
the  theoretic  velocity  is  uniform  over  the  section.  For 
this  reason  the  coefficients  of  discharge  probably  vary 
less  with  the  head  than  for  the  previous  cases. 

Submerged  orifices  are  used  for  canal-locks,  tide-gates, 
filter-beds,  for  the  discharge  of  waste  water  through  dams, 
and  for  the  admission  of  w^ater  from  a  canal  to  a  power- 
plant.  The  inner  edges  of  such  orifices  are  usually  rounded 
so  that  the  contraction  is  suppressed,  and  the  coefficient 
of  discharge  may  then  be  higher  than  0.9  (Art.  54). 

Prob.  52.  An  orifice  one  inch  square  in  a  gate  such  as  shown 
in  Fig.  19a,  Art.  19,  is  3  feet  below  the  higher  water  level  and 
2  feet  below  the  lower  water  level.  Compute  the  discharge  in 
cubic  feet  per  second,  and  also  in  gallons  per  minute. 


ART.  53.     SUPPRESSION  OF  THE  CONTRACTION 

When  a  vertical  orifice  has  its  lower  edge  at  the  bottom 
of  the  reservoir,  as  shown  at  A  in  Fig.  53,  the  particles 
of  water  flowing  through  its  lower  por- 
tion move  in  lines  nearly  perpendicular 
to  the  plane  of  the  orifice,  or  the  con- 
traction of  the  jet  does  not  form  on 
the  lower  side.  This  is  called  a  case  of 
suppressed  or  incomplete  contraction. 
The  same  thing  occurs,  but  in  a  lesser 
degree,  when  the  lower  edge  of  the  orifice  is  near  the  bottom 
as  shown  at  B.  In  like  manner,  if  an  orifice  be  placed 
so  that  one  of  its  vertical  edges  is  at  or  near  a  side  of  the 
reservoir,  as  at  C,  the  contraction  of  the  jet  is  suppressed 
upon  one  side,  and  if  it  be  placed  at  the  lower  corner 
of  the  reservoir  suppression  occurs  both  upon  one  side 
and  the  lower  part  of  the  jet. 


ART.  53  SUPPRESSION    OF    THE    CONTRACTION  129 

The  effect  of  suppressing  the  contraction  is,  of  course, 
to  increase  the  cross-section  of  the  jet  at  the  place  where 
full  contraction  would  otherwise  occur,  and  it  is  found 
by  experiment  that  the  discharge  is  likewise  increased. 
Experiments  also  show  that  more  or  less  suppression'  of 
the  contraction  will  occur  unless  each  edge  of  the  orifice 
is  at  a  distance  at  least  equal  to  three  times  its  least  diameter 
from  the  sides  or  bottom  of  the  reservoir. 

The  experiments  of  Lesbros  and  Bidone  furnish  the 
means  of  estimating  the  increased  discharge  caused  by 
suppression  of  the  contraction.  They  indicate  that  for 
square  orifices  with  contraction  suppressed  on  one  side 
the  coefficient  of  discharge  is  increased  about  3.5  percent, 
and  with  contraction  suppressed  on  two  sides  about  7.5 
percent.  For  a  rectangular  orifice  with  the  contraction 
suppressed  on  the  bottom  edge  the  percentages  are  larger, 
being  about  6  or  7  percent  when  the  length  of  the  rectangle 
is  four  times  its  height,  and  from  8  to  12  percent  when 
the  length  is  twenty  times  the  height.  The  percentage 
of  increase,  moreover,  varies  with  the  head,  the  lowest 
heads  giving  the  lowest  percentages. 

It  is  apparent  that  suppression  of  the  contraction 
should  be  avoided  if  accurate  results  are  desired.  The 
experiments  from  which  the  above  conclusions  are  deduced 
were  made  upon  small  orifices  with  heads  less  than  6 
feet,  and  it  is  not  known  how  they  will  apply  to  large 
orifices  under  high  heads.  For  a  rectangular  orifice  of 
length  about  three  times  its  height,  with  contraction  sup- 
pressed on  the  ends  and  bottom,  the  coefficient  of  discharge 
is  probably  about  0.75. 

Prob.  53a.  Compute  the  probable  discharge  from  a  vertical 
orifice  one  foot  square  when  the  head  on  its  upper  edge  is  4  feet, 
the  contraction  being  suppressed  on  the  lower  edge. 

Prob.  536.  Compute  the  discharge  for  the  same  data  when 
contraction  is  suppressed  on  all  sides. 


130 


FLOW    THROUGH    ORIFICES 


CHAP.  V 


ART.  54.     ORIFICES  WITH  ROUNDED  EDGES 

If  the  inner  edge  of  the  orifice  be  made  rounded,  as 
shown  in  Fig.  54,  the  contraction  of  the  jet  is  modified, 

and  the  discharge  is  increased. 
With  a  slight  degree  of  rounding, 
as  at  A,  a  partial  contraction 
occurs;  but  with  a  more  complete 
bounding,  as  at  C,  the  particles 
o  of  water  issue  perpendicular  to 
the  plane  of  the  orifice  and  there 
is  no  contraction  of  the  jet.  If  a 
be  the  area  of  the  least  cross- 
section  of  the  orifice,  and  a'  that  of  the  jet,  the  coefficient 
of  contraction  as  defined  in  Art.  44  is 


c'=af/a 


(54) 


For  a  standard  orifice  with  sharp  inner  edges  (Art.  43) 
the  mean  value  of  cf  is  0.62,  but  for  an  orifice  with  rounded 
edges,  c'  may  have  any  value  between  0.62  and  i.o,  de- 
pending upon  the  degree  of  rounding. 

The  coefficient  of  discharge  c  for  standard  orifices  has 
a  mean  value  of  about  0.6 1 ;  this  is  increased  with  rounded 
edges  and  may  have  any  value  between  0.61  and  i.o. 
A  rounded  interior  edge  in  an  orifice  is  therefore  always 
a  source  of  error  when  the  object  of  the  orifice  is  the 
measurement  £>f  the  discharge.  If  a  contract  provides 
that  water  shall  be  gaged  by  standard  orifices,  care  should 
always  be  taken  that  the  interior  edges  do  not  become 
rounded  either  by  accident  or  by  design. 

Prob.  54a.  Compute  the  discharge  from  an  orifice  4  inches 
in  diameter  under  a  head  of  6  feet,  when  £=0.89. 

Prob.  546.  If  an  orifice  with  rounded  edges  has  a  coefficient 
of  velocity  of  0.88  and  a  coefficient  of  discharge  of  0.75,  find 
the  coefficient  of  contraction. 


ART.  55  WATER   MEASUREMENT   BY   ORIFICES  131 


ART.  55.     WATER  MEASUREMENT  BY  ORIFICES 

In  order  that  water  may  be  accurately  measured  by 
the  use  of  orifices  many  precautions  must  be  taken,  some 
of  which  have  already  been  noted,  but  may  here  be  briefly 
recapitulated.  The  area  of  the  orifice  should  be  small 
compared  with  the  size  of  the  reservoir  in  order  that 
velocity  of  approach  may  not  exist,  or  if  this  cannot  be 
avoided  it  should  be  taken  into  account  by  formula  (51)!. 
The  inner  edge  of  the  orifice  must  have  a  definite  right- 
angled  corner,  and  its  dimensions  are  to  be  accurately 
determined.  If  the  orifice  be  in  wood,  care  should  be 
taken  that  the  inner  surface  be  smooth,  and  that  it  be 
kept  free  from  the  slime  which  often  accompanies  the 
flow  of  water  even  when  apparently  clear.  That  no 
suppression  of  the  contraction  may  occur,  the  edges  of 
the  orifice  should  not  be  nearer  than  three  times  its  least 
dimension  to  a  side  of  the  reservoir. 

Orifices  under  very  low  heads  should  be  avoided, 
because  slight  variations  in  the  head  produce  relatively 
large  errors,  and  also  because  the  coefficients  of  discharge 
vary  more  rapidly  and  are  probably  not  so  well  determined 
as  for  cases  where  the  head  is  greater  than  four  times  the 
depth.  If  the  head  be  very  low  on  an  orifice,  vortices 
will  form  which  render  any  estimation  of  the  discharge 
unreliable. 

The  measurement  of  the  head,  if  required  with  pre- 
cision, must  be  made  with  the  hook  gage  described  in 
Art.  35.  For  heads  greater  than  two  or  three  feet  the 
readings  of  an  ordinary  glass  gage  placed  upon  the  out- 
side of  the  reservoir  will  usually  prove  sufficient,  as  this 
can  be  read  to  hundredths  of  a  foot  with  accuracy.  An 
error  of  o.oi  feet  when  the  head  is  3.00  feet  produces  an 
error  in  the  computed  discharge  of  less  than  two-tenths 
of  one  percent;  for,  the  discharges  being  proportional 


132  FLOW  THROUGH  ORIFICES  CHAP,  v 

to  the  square  roots  of  the  heads,  the  square  root  of  3.01 
divided  by  the  square  root  of  3.00  equals  1.0017.  For 
the  rude  measurements  in  connection  with  the  miner's 
inch  a  common  foot-rule  will  usually  suffice. 

The  effect  of  temperature  upon  the  discharge  remains 
to  be  noticed;  this  is  only  appreciable  with  small  orifices 
and  under  low  heads  and  hence  such  orifices  and  heads 
are  not  desirable  in  precise  measurements.  Unwin  found 
that  the  discharge  was  diminished  one  percent  by  a  rise 
of  144  degrees  in  temperature;  his  orifice  was  a  circle 
0.033  fe^  *n  diameter  under  heads  ranging  from  i.o  to 
1.5  feet.  Smith  found  that  the  discharge  was  diminished 
one  percent  by  a  rise  of  55  degrees  in  temperature;  his 
orifice  was  a  circle  0.02  feet  in  diameter  under  heads  rang- 
ing from  0.56  to  3.2  feet. 

The  coefficients  given  in  Tables  17-22  may  be  supposed 
liable  to  a  probable  error  of  about  two  units  in  the  third 
decimal  place:  thus  a  coefficient  0.615  should  really  be 
written  o.6i5±o.oo2;  that  is,  the  actual  value  is  as  likely 
to  be  between  0.613  and  0.617  as  to  be  outside  of  those 
limits.  The  probable  error  in  computed  discharges  due 
to  the  coefficient  is  hence  nearly  one-half  of  one  percent. 
To  this  are  added  the  errors  due  to  inaccuracy  of  observa- 
tion, so  that  it  is  thought  that  the  probable  error  of  care- 
ful work  with  standard  circular  orifices  is  at  least  one 
percent.  The  computed  discharges  are  hence  liable  to 
error  in  the  third  significant  figure,  so  that  it  is  useless 
to  carry  numerical  results  beyond  three  figures  when 
based  upon  tabular  coefficients.  As  a  precise  method 
of  measuring  small  quantities  of  water,  standard  orifices 
take  a  high  rank  when  the  observations  are  conducted 
with  care. 

Prob.  55.  If  e  be  a  small  error  in  measuring  the  head  hf 
show  that  the  error  in  the  computed  discharge  q  due  to  this 
cause  is  qe/zh. 


ART.  56  LOSS    OF   ENERGY    OR   HEAD  133 


ART.  56.     Loss  OF  ENERGY  OR  HEAD 

A  jet  of  water  flowing  from  an  orifice  possesses  by 
virtue  of  its  velocity  a  certain  kinetic  energy,  which  is 
always  less  than  the  theoretic  potential  energy  due  to 
the  head  (Art.  28).  Let  h  be  the  head  and  W  the  weight 
of  water  discharged  per  second,  then  the  theoretic  energy 
per  second  is 


Let  v  be  the  actual  velocity  of  the  water  at  the  contracted 
section  of  the  jet  ;  then  the  actual  energy  per  second  of  the 
water  as  it  passes  that  section  is 

k.-W.v*/ag 

Now  let  ct  be  the  coefficient  of  velocity  (Art.  45)  ;  then 
v2  =  cl2.2gh,  and  accordingly  the  actual  energy  of  the  jet 
per  second  is 


The  efficiency  of  the  jet,  or  the  ratio  of  the  actual  to  the 
theoretic  energy  now  is 

e  =  k/K  =  c*  (56) 

which  is  a  number  always  less  than  unity. 

For  the  standard  orifice  the  mean  value  of  cl  is  0.98, 
and  hence  a  mean  value  of  c^2  is  0.96.  The  actual  energy 
of  a  jet  from  such  an  orifice  is  hence  about  96  percent 
of  the  theoretic  energy,  and  the  loss  of  energy  is  about 
4  percent.  This  loss  is  due  to  the  fractional  resistance 
of  the  edges  of  the  orifice,  whereby  the  energy  of  pressure 
or  velocity  is  changed  into  heat. 

In  the  plane  of  the  standard  orifice  the  velocity  is 
slower  than  at  the  contracted  section  since  the  area  there 
is  greater.  If  vl  be  this  velocity,  a  the  area  of  the  orifice, 
and  a'  that  of  the  jet  at  the  contracted  section,  it  is  clear 
that  Q,VI  =  a'v  or  v^  =  c'v,  where  c'  is  the  coefficient  of  con- 


134  PLOW  THROUGH  ORIFICES  CHAP,  v 

traction  0.62.  The  kinetic  energy  in  the  plane  of  the 
orifice  is  W.v1*/2g,  or  o.^Wvz/2g,  or  o.tfWh.  Thus, 
in  the  plane  of  the  orifice  4  percent  of  the  theoretic  energy 
is  lost  overcoming  friction,  37  percent  is  in  the  form  of 
kinetic  energy,  and  the  remaining  59  percent  exists  in 
the  form  of  pressure  energy.  This  59  percent  is  trans- 
formed into  kinetic  energy  when  the  water  has  reached 
the  contracted  section. 

In  hydraulics  the  terms  energy  and  head  are  often 
used  as  synonymous,  although  really  energy  is  proportional 
to  head.  Thus  the  pressure-head  that  causes  the  flow 
is  h  and  the  velocity-head  of  the  issuing  jet  is  vz/2g,  and 
these  are  proportional  to  the  theoretic  and  effective  en- 
ergies. The  lost  head  h'  is  the  difference  of  these,  or 


and  this  applies  not  only  to  an  orifice  but  to  any  tube 
or  pipe.     Inserting  for  v2  its  value  this  becomes 


which  gives  the  lost  head  in  terms  of  the  total  head.     In- 
serting for  h  its  value  in  terms  of  v  reduces  this  to 


which  gives  the  lost  head  in  terms  of  the  velocity-head. 
Thus,  for  an  orifice  whose  coefficient  of  velocity  is  0.97 
the  lost  head  h'  is  o.o6oh  or  o.o6$v2/2g.  For  the  standard 
orifice  the  lost  head  h'  is  0.040/2.  or  o.o4iv2/2g.  When 
velocity  of  approach  exists  the  value  of  h  is  to  be  increased 
by  the  velocity-head  of  the  approaching  water. 

Prob.  56.  Water  approaches  an  orifice  with  a  velocity  of 
1.5  feet  per  second,  the  pressure-head  on  the  orifice  is  2.435  feet, 
and  the  coefficient  of  velocity  is  0.975.  Compute  the  loss  of 
head. 


ART.  57  DISCHARGE    UNDER   A    DROPPING    HEAD  135 


ART.  57.     DISCHARGE  UNDER  A  DROPPING  HEAD 

If  a  vessel  or  reservoir  receives  no  inflow  of  water 
while  an  orifice  is  open,  the  head  drops  and  the  discharge 
decreases  in  each  successive  second.  Let  H  be  the  head 
on  the  orifice  at  a  certain  instant,  and  h  the  head  t  seconds 
later;  let  A  be  the  area  of  the  uniform  horizontal  cross- 
section  of  the  vessel,  and  a  the  area  of  the  orifice.  Then, 
the  theoretic  time  t  is  given  by  the  second  formula  in 
Art.  26.  To  determine  the  actual  time  the  coefficient  of 
discharge  must  be  introduced.  Referring  to  the  demon- 
stration, it  is  seen  that  a*V2gy.  dt  is  the  theoretic  discharge 
in  the  time  dt]  hence  the  actual  discharge  is  c.a\/2gydt, 
and  accordingly  a  in  the  above-mentioned  formula  is  to  be 
replaced  by  ca,  or 

(57), 


cav  2g 

is  the  practical  formula  for  the  time  in  which  the  water 
level  drops  from  H  to  h.  In  using  this  formula  c  is  to 
be  taken  from  the  tables  at  the  end  of  this  volume,  an 
average  value  being  selected  corresponding  to  the  average 
head. 

Experiments  have  been  made  to  determine  the  value 
of  c  by  the  help  of  this  formula;  the  liquid  being  allowed 
to  flow,  A,  a,  H,  ht  and  t  being  observed,  whence  c  is  com- 
puted. In  this  way  c  for  mercury  has  been  found  to  be 
about  0.62.*  Only  approximate  mean  values  can  be 
found  in  this  manner,  since  c  varies  with  the  head,  par- 
ticularly for  small  orifices  (Art.  47).  For  a  large  orifice 
the  time  of  descent  is  usually  so  small  that  it  cannot  be 
noted  with  precision,  and  the  friction  of  the  liquid  on 
the  sides  of  the  vessel  may  also  introduce  an  element 
of  uncertainty.  Further,  if  h  be  small  a  vortex  forms 

*  Downing's  Elements  of  Practical  Hydraulics  (London,  1875),  p.  187. 


136  FLOW  THROUGH  ORIFICES  CHAP,  v 

which  renders  the  formula  unreliable.  This  experiment 
has  therefore  little  value  except  as  illustrating  and  con- 
firming the  truth  of  the  theoretic  formulas. 

The  discharge  in  one  second  when  the  head  is  H  at 
the  beginning  of  that  second  is  found  as  follows:  The 
above  equation  may  be  written  in  the  form 

VJT-  tca\/2g/2A  =  Vh 

By  squaring  both  members,  transposing,  and  multiplying 
by  A,  this  may  be  reduced  to 

A(H-h)= 

But  the  first  member  of  this  equation  is  the  quantity 
discharged  in  t  seconds;  therefore  the  discharge  in  the 
first  second  is 

q  =  ca\/2g(VH  -  ca\/2g/4A) 

If  A  =  oo ,  this  becomes  cay2gH,  which  should  be  the 
case,  for  then  H  would  remain  constant.  At  the  end  of 
the  first  second  the  water  level  has  fallen  the  amount 
q/A,  so  that  the  head  at  the  beginning  of  the  second 
second  is  H  —  q/A. 

For  example,  let  an  orifice  one  foot  square  in  a  reservoir 
of  10  square  feet  section  be  under  a  head  of  9  feet,  and 
c  =  0.602.  Then  the  discharge  in  one  second  is  13.9  cubic 
feet,  and  the  head  drops  to  7.61  feet.  The  discharge  in 
the  next  second  is  12.7  cubic  feet,  and  the  head  drops  to 
6.34  feet. 

Prob.  57 a.  For  the  data  of  the  last  paragraph  compute  the 
number  of  seconds  required  to  lower  the  head  of  9  feet  down 
to  3  feet. 

Prob.  576.  Find  the  time  required  to  discharge  480  gallons 
of  water  from  an  orifice  2  inches  in  diameter  at  8  feet  below  the 
*vater  level  when  the  cross-section  of  the  tank  is  4X4  feet. 


ART.  58  EMPTYING   AND    FILLING   A   CANAL   LOCK  137 


ART.  58.     EMPTYING  AND  FILLING  A  CANAL  LOCK 

A  canal  lock  is  emptied  by  opening  one  or  more  orifices 
in  the  lower  gates.  Let  a  be  their  area  and  H  the  head 
of  water  on  them  when  the  lock  is  full;  let  A  be  the  area 
of  the  horizontal  cross-section  of  the  lock.  Then  in  the 
first  formula  of  the  last  article  h=o,  and  the  time  of 
emptying  the  lock  is 

t  =  2  A  VW/caVlzg  (58) 

If  the  discharge  be  free  into  the  air,  H  is  the  distance 
from  the  center  of  the  orifice  to  the  level  of  the  water  in 
the  lock  when  filled;  but  if,  as  is  usually  the  case,  the 
orifices  be  below  the  level  of  the  water  in  the  tail  bay, 
H  is  the  difference  in  height  between  the  two  water  levels. 
The  tail  bay  is  regarded  as  so  large  compared  with  the 
lock  that  its  water  level  remains  constant  during  the  time 
of  emptying. 

For  example,  let  it  be  required  to  find  the  time  ofp 
emptying  a  canal  lock  80  feet  long  and  20  feet  wide  through 
two  orifices  each  of  4  square  feet  area,  the  head  upon 
which  is  1 6  feet  when  the  lock  is  filled.  Using  for  c  the 
value  0.6  for  orifices  with  square  inner  edges,  the  formula 
gives 

2X80X20X4 
*  =   0.6X8X8.02    =  333  Sec°nds  =  5*  ^nutes 

If,  however,  the  circumstances  be  such  that  c  is  0.8,  the 
time  is  about  250  seconds,  or  4^  minutes.  It  is  therefore 
seen  that  it  is  important  to  arrange  the  orifices  of  discharge 
in  canal  locks  with  rounded  inner  edges. 

The  filling  of  the  lock  is  the  reverse  operation.  Here 
the  water  in  the  head  bay  remains  at  a  constant  level 
and  the  discharge  through  the  orifices  in  the  upper  gates 
decreases  with  the  rising  head  in  the  lock.  Let  H  be  the 
effective  head  on  the  orifices  when  the  lock  is  empty, 


138 


FLOW   THROUGH    ORIFICES 


CHAP.  V 


and  y  the  effective  head  at  any  time  t  after  the  beginning 
of  the  discharge.  The  area  of  the  section  of  the  lock 
being  At  the  quantity  Ady  is  discharged  in  the  time  dt, 


Head  Bay_~^r  ITr"  ~.~  12  zrzzn= 


FIG.  28 


and  this  is  equal  to  ca\r*gy  dt,  if  a  be  the  area  of  the 
orifices  and  c  the  coefficient  of  discharge.  Hence  the  same 
expression  as  (58)  results,  and  the  times  of  filling  and 
emptying  a  lock  are  equal  if  the  orifices  are  of  the  same 
dimensions  and  under  the  same  heads.  The  area  required 
for  the  orifices  may  be  found  for  any  case  from  (58)  when 
A,  H,  t,  and  c  are  given. 

Prob.  58a.  Compute  the  areas  of  the  two  orifices  when 
A  =  1800  square  feet,  2  =  3  minutes,  ^  =  0.7,  #  =  7  feet  for  the 
upper  and  12  feet  for  the  lower  orifice. 

Prob.  586.  A  lock  90  feet  long  and  20  feet  wide,  with  a  lift 
of  12  feet,  contains  a  boat  weighing  500  net  tons.  When  the 
lock  is  emptied  in  order  to  lower  the  boat,  how  much  water 
flows  from  the  lower  orifices?  If  the  cross-section  of  these 
orifices  is  12.3  square  feet  and  £  =  0.7,  what  is  the  time  of 
emptying? 

ART.  59.     COMPUTATIONS  IN  METRIC  MEASURES 

Most  of  the  formulas  of  this  chapter  are  rational  and 
may  be  used  in  all.  systems  of  measures.  The  coefficients 
of  contraction,  velocity,  and  discharge  are  abstract  num- 
bers, which  are  the  same  in  all  systems,  like  the  constants 
of  mathematics.  In  the  metric  system  the  area  a  is  to 


ART.  59  COMPUTATIONS    IN    METRIC    MEASURES  -   139 

be  taken  in  square  meters,  the  head  h  in  meters,  \/2g 
as  4.427,  and  then  the  discharge  q  willbe  in  cubic  meters 
per  second. 

(Art.  47)  For  standard  circular  vertical  orifices  the 
formulas  (47)  t  and  (47) 2  apply  to  the  metric  system  if 
8.02  be  replaced  by  4.427.  In  using  these  the  coefficient 
c  may  be  taken  from  Table  18  which  has  been  adapted 
to  metric  arguments  from  Table  17.  For  example,  if 
the  diameter  of  the  orifice  be  2.5  centimeters  and  the  head 
on  its  center  be  0.6  meters,  interpolation  in  the  table 
gives  the  value  of  c  as  0.606. 

(Art.  48)  For  standard  square  vertical  orifices^  the 
formulas  (48) t  and  (48) 2  are  changed  to  the  metric  system 
by  substituting  4.427  for  8.02  and  2.951  for  5.347.  Table 
20  gives  values  of  the  coefficient  c  for  arguments  in  metric 
measures. 

(Art.  49)  Table  21  has  not  been  transformed  into 
one  with  metric  arguments,  as  it  applies  only  to  the  special 
case  where  the  rectangular  orifice  is  one  foot  wide.  If 
the  heads  in  the  first  column  be  changed  into  meters, 
by  writing  0.12  meters  for  0.4  feet,  0.18  meters  for  0.6 
feet,  etc.,  and  the  numbers  at  the  top  be  changed  into 
centimeters  by  writing  3.8  centimeters  for  0.125  feet, 
7.6  centimeters  for  0.25  feet,  etc.,  the  table  will  be  ready 
for  use  with  metric  arguments  for  rectangular  orifices  30.5 
centimeters  wide. 

(Art.  50)  The  miner's  inch,  when  the  head  on  the 
center  of  the  orifice  is  16.5  centimeters,  is  0.0433  cubic 
meters  or  43.3  liters  per  minute. 

(Art.  58)  In  using  (58)  in  the  metric  system,  a  and 
A  are  to  be  taken  in  square  meters,  H  in  meters,  g  as 
9.80  meters  per  second  per  second,  and  \/2g  as  4.427; 
q  will  then  be  found  in  cubic  meters. 

Prob.  59a.  Michelotti  found  the  range  of  a  jet  to  be  6.25 
meters' on  a  horizontal  plane  1.41  meters  below  the  vertical 


140  '  FLOW   THROUGH    ORIFICES  CHAP.  V 

orifice,  which  was  under  a  head  of  7.19  meters.  Compute  the 
coefficient  of  velocity. 

Prob.  596.  An  orifice  3  centimeters  square  was  under  a 
constant  head  of  4  meters,  and  during  230  seconds  the  jet 
flowed  into  a  tank  which  was  found  to  contain  112.2  liters. 
Show  that  the  coefficient  of  discharge  was  0.612. 

Prob.  59c.  Find  from  the  table  the  coefficient  of  discharge 
for  a  standard  circular  orifice  2.5  centimeters  in  diameter 
under  a  head  of  2.5  meters. 

Prob.  59 d.  Compute  the  discharge  through  a  standard 
orifice  7.5  centimeters  square  under  a  head  of  8  meters. 

Prob.  59e.  Compute  the  time  required  to  empty  a  canal 
lock  7  meters  wide  and  32  meters  long  through  an  orifice  of 
0.9  square  meters  area,  the  head  on  the  center  of  the  orifice 
being  5.1  meters  when  the  lock  is  filled. 


ART.  60 


DESCRIPTION  OF  WEIRS 


141 


CHAPTER  VI 
FLOW  OF  WATER  OVER  WEIRS 

ART.  60.     DESCRIPTION  OF  WEIRS 

A  weir  is  a  notch  in  the  top  of  the  vertical  side  of  a 
vessel  or  reservoir  through  which  water  flows.  The  notch 
is  generally  rectangular,  and  the  word  weir  will  be  used 
to  designate  a  rectangular  notch  unless  otherwise  speci- 
fied, the  lower  edge  of  the  rectangle  being  truly  horizontal, 
and  its  sides  vertical.  The  lower  edge  of  the  rectangle 
is  called  the  "crest"  of  the  weir.  In  Fig.  60a  is  shown 
the  outline  of  the  most  usual  form,  where  the  vertical 
edges  of  the  notch  are  sufficiently  removed  from  the  sides 


FIG.  60a 


FIG.  606 


of  the  reservoir  or  feeding  canal,  so  that  the  sides  of  the 
stream  may  be  fully  contracted;  this  is  called  a  weir  with 
end  contractions.  In  the  form  of  Fig.  606  the  edges  of 
the  notch  are  coincident  with  the  sides  of  the  feeding 
canal,  so  that  the  filaments  of  water  along  the  sides  pass 
over  without  being  deflected  from  the  vertical  planes  in 


142  FLOW  OVER  WEIRS  CHAP,  vi 

which  they  move;  this  is  called  a  weir  without  end  con- 
tractions, or  with  end  contractions  suppressed.  Both  kinds 
of  weirs  are  extensively  used  for  the  measurement  of  wrater 
in  engineering  operations. 

It  is  necessary  in  order  to  make  accurate  measurements 
of  discharge  by  a  weir  that  the  same  precaution  should 
be  taken  as  for  orifices  (Art.  55),  namely,  that  the  inner 
edge  of  the-  notch  shall  be  a  definite  angular  corner  so 
that  the  water  in  flowing  out  may  touch  the  crest  only 
in  a  line,  thus  insuring  complete  contraction.  In  precise 
observations  a  thin  metal  plate  will  be 
used  for  a  crest,  while  in  common 
work  it  may  be  sufficient  to  have  the 
crest  formed  by  a  plank  of  smooth 
hard  wood  with  its  inner  corner  cut- 
FIG.  60c  to  a  sharp  right  angle  and  its  outer 

edge  beveled.  The  vertical  edges  of  the  weir  should  be 
made  in  the  same  manner  for  weirs  with  end  contrac- 
tions, while  for  those  without  end  contractions  the  sides 
of  the  feeding  canal  should  be  smooth  and  be  prolonged 
a  slight  distance  beyond  the  crest.  It  is  also  necessary 
to  observe  the  same  precautions  as  for  orifices  to  prevent 
the  suppression  of  the  contraction  (Art.  53),  namely,  that 
the  distance  from  the  crest  of  the  weir  to  the  bottom  of 
the  feeding  canal,  or  reservoir,  should  be  greater  than  three 
times  the  head  of  water  on  the  crest.  For  a  weir  with  end 
contractions  a  similar  distance  should  exist  between  the 
vertical  edges  of  the  weir  and  the  sides  of  the  feeding 
canal. 

The  head  of  water  H  upon  the  crest  of  a  weir  is  usually 
much  less  than  the  breadth  of  the  crest  b.  The  value 
of  H  should  not  be  less  than  o.i  foot,  and  it  rarely  ex- 
ceeds 1.5  feet.  The  least  value  of  b  in  practice  is  about 
0.5  feet,  and  it  does  not  often  exceed  20  feet.  Weirs 
are  extensively  used  for  measuring  the  discharge  of  small 


ART.  60  DESCRIPTION  OF  WEIRS  143 

streams,  and  for  determining  the  quantity  of  water  sup- 
plied to  hydraulic  motors;  the  practical  importance  of 
the  subject  is  so  great  that  numerous  experiments  have 
been  made  to  ascertain  the  laws  of  flow,  and  the  coefficients 
of  discharge. 

Since  the  head  on  the  crest  of  a  weir  is  small,  it  must 
be  determined  with  precision  in  order  to  avoid  error  in 
the  computed  discharge.  The  hook  gage  which  is  illustrated 
in  Art.  35  is  generally  used  for  accurate  work  in  connection 
with  hydraulic  motors,  and  the  simpler  form,  consisting 
of  a  hook  set  into  a  leveling  rod,  is  usually  of  sufficient 
precision  for  many  cases.  For  rough  gagings  of  streams 
the  heads  may  be  determined  by  setting  a  post  a  few 
feet  up-stream  from  the  weir  and  on  the  same  level  as 
the  crest,  and  measuring  the  depth  of  the  water  over  the 
top  of  the  post  by  a  scale  graduated  to  tenths  and  hun- 
dredths  of  a  foot,  the  thousandths  being*  either  estimated 
or  omitted  entirely. 

The  head  H  on  the  crest  of  the  weir  is  in  all  cases  to 
be  measured  several  feet  up-stream  from  the  crest,  as 
indicated  in  Fig.  60c.  This  is  necessary  because  of  the 
curve  taken  by  the  surface  of  the  water  in  approaching 
the  weir.  The  distance  to  which  this  curve  extends  back 
from  the  crest  of  the  weir  depends  upon  many  circum- 
stances (Art.  69),  but  it  is  generally  considered  that 
perfectly  level  water  will  be  found  at  2  or  3  feet  back 
of  the  crest  for  small  weirs,  and  at  6  or  8  feet  for 
very  large  weirs.  It  is  desirable  that  the  hook  should 
be  placed  at  least  one  foot  from  the  sides  of  the  feeding 
canal,  if  possible.  As  this  is  apt  to  render  the  position 
of  the  observer  uncomfortable,  some  experimenters  have 
placed  the  hook  in  a  pail  a  few  feet  away  from  the  canal, 
the  water  being  led  to  the  pail  by  a  pipe  which  joins  the 
feeding  canal  several  feet  back  from  the  crest,  and  the 
water  should  enter  this  pipe,  not  at  its  end,  but  through  a 
a  number  of  holes  drilled  at  intervals  along  its  circumfer- 


144  FLOW  OVER  WEIRS  CHAP,  vi 

ence.  Piezometers  (Art.  36)  consisting  of  a  glass  tube  and 
scale  are  also  sometimes  used  for  large  heads,  the  water 
being  led  to  the  tube  by  such  a  pipe. 

Prob.  60.  The  trough  of  a  weir,  several  feet  back  from  the 
crest,  is  4.0  feet  wide,  and  the  depth  of  water  in  it  is  1.96  feet. 
What  is  the  mean  velocity  in  this  trough  when  the  flow  over 
the  weir  is  4.24  cubic  feet  per  second? 

ART.  61.     FORMULAS  FOR  DISCHARGE 

Referring  to  the  demonstration  of  Art.  48  it  is  seen 
that  a  rectangular  orifice  becomes  a  weir  when  the  head 
on  its  top  is  zero.  Let  b  be  the  breadth  of  the  notch, 
commonly  called  the  length  of  the  crest,  and  H  thejiead 
of  water  -QQjbhe  crest.  Then  replacing  h±  by  o  and  h2  by 
H,  the  theoretic  discharge  per  second  is 


(61), 

The  head  H  is  not  the  depth  measured  in  the  plane  of 
the  crest,  for  since  the  deduction  of  the  formula  assumes 
nothing  regarding  the  fall  due  to  the  surface  curve,  and 
regards  the  velocity  at  any  point  vertically  over  the  crest 
as  due  to  the  head  upon  that  point  below  the  free  water 
surface,  it  seems  that  H  should  be  measured  with  reference 
to  that  surface,  as  is  actually  done  by  the  hook  gage. 
The  above  formula  then  gives  the  theoretic  discharge  per 
second,  provided  that  there  be  no  velocity  at  the  point 
where  H  is  measured,  which  can  only  be  the  case  when 
the  area  of  the  weir  opening  is  very  small  compared  to 
that  of  the  cross-section  of  the  feeding  canal.  This  con- 
dition would  be  fulfilled  for  a  rectangular  notch  placed 
at  the  side  of  a  large  pond. 

When  there  is  an  appreciable  velocity  of  approach 
of  the  water  at  the  point  where  H  is  measured  by  the 
hook  gage,  the  above  formula  must  be  modified.  Let  v 
be  the  mean  velocity  in  the  feeding  canal  at  this  section; 


ART.  61  FORMULAS  FOR  DISCHARGE  145 

this  velocity  may  be  regarded  as  due  to  a  fall,  h,  from 

the    surface    of    still    water    at 

some   distance   up-stream   from 

the  hook,  as  shown  in  Fig.  61. 

Now  the  true  head  on  the  crest 

of   the    weir    is    H  +  h,    as    this 

would    have   -been    the    reading  FIG  61 

of  the  hook  gage  had  it  been 

placed  where  the  water  had  no  velocity.     Accordingly  the 

theoretic  discharge  per  second  is 


in  which  H  is  read  by  the  hook  and  h  is  to  be  determined 
from  the  mean  velocity  v. 

The  actual  discharge  is  always  less  than  the  theoretic 
discharge,  due  to  the  contraction  of  the  stream  and  the 
resistances  of  the  edges  of  the  weir.  To  take  account 
of  these  a  coefficient  is  applied  to  the  theoretic  formulas 
in  the  same  manner  as  for  orifices;  these  coefficients  be- 
ing determined  by  experiment,  the  formulas  may  then 
be  used  for  cornputing  the  actual  discharge.  It  has  also 
been  proposed  by  Hamilton  Smith  to  modify  the  head 
h,  owing  to  the  fact  that  the  velocity  of  approach  is  not 
constant  throughout  the  section,  but  greater  near  the 
surface  than  near  the  bottom,  as  in  conduits  and  streams. 
(Art.  118).  Accordingly  the  following  may  be  written  as 
an  expression  for  the  actual  discharge  : 

q  =  c.%V^.b(H  +  nh)*  ^T  (61)2 

in  which  c  is  the  coefficient  of  discharge  whose  value  is 
always  less  than  unity,  and  n  is  a  number  which  lies  be- 
tween i.o  and  1.5.  For  the  English  system  of  measures 
a  mean  value  of  \/2g  is  8.020,  but  a  more  precise  value 
can  be  computed  from  (7)!  for  any  locality. 

The  above  formulas  are  not  in  all  respects  perfectly 
satisfactory,  and  indeed  many  others  have  been  proposed, 


146  FLOW  OVER  WEIRS  CHAP,  vi 

one  of  these  being  derived  from  (51  )4  by  making  h0=h, 
h2=H,  and  7i1=o.  The  actual  discharge  differs,  however, 
so  much  from  the  theoretical  that  the  final  dependence 
must  be  upon  the  coefficients  deduced  from  experiment, 
and  hence  any  fairly  reasonable  formula  may  be  used 
within  the  limits  for  which  its  coefficients  have  been 
established.  In  spite  of  the  objections  which  may  be 
raised  against  all  forms  of  formulas,  the  fact  remains 
that  the  measurement  of  water  by  weirs  is  one  of  the  most 
convenient  methods,  and  probably  the  most  precise  method, 
unless  the  quantity  is  so  small  as  to  pass  through  a  circular 
orifice  less  than  one  foot  in  diameter.  With  proper  pre- 
cautions the  probable  error  in  measurements  of  discharge- 
by  weirs  should  be  less  than  two  or  three  percent. 

Prob.  61a.  Find  the  velocity-head  h  when  the  mean  velocity 
of  approach  is  20  feet  per  minute. 

Prob.  616.  Show  by  using  formula  (61)  that  an  error  of 
about  one-half  of  one  percent  results  in  the  computed  discharge 
if  an  error  of  o.ooi  feet  be  made  in  reading  the  head  when. 
#=0.3  feet. 

% 
ART.  62.     VELOCITY  OF  APPROACH 

The  head  h  which  produces  the  velocity  v  is  expressed 
by  v*/2g,  and  in  the  case  of  a  weir,  the  velocity  of  approach 
v  is  due  to  a  fall  from  the  height  h\  thus  the  velocity- 
head  is 

h  = 


and  when  v  is  known  h  can  be  computed.  One  way  of 
finding  v  is  to  observe  the  time  of  passage  of  a  float  through 
a  given  distance;  but  this  is  not  a  precise  method.  The 
usual  method  is  to  compute  v  from  an  approximate  value 
of  the  discharge,  which  is  first  computed  by  regarding  v, 
and  hence  h,  as  zero.  This  determination  is  rendered 
possible  by  the  fact  that  v  is  usually  small,  and  hence 
that  h  is  quite  small  as  compared  with  H. 


ART.  62  VELOCITY  OF  APPROACH  147 

Let  B  be  the  breadth  of  the  cross-section  of  the  feeding 
canal  at  the  place  where  the  readings  of  the  hook  are 
taken,  and  let  G  be  its  depth  below  the  crest  (Fig.  (61). 
The  area  of  that  cross-section  then  is 


The  mean  velocity  in  this  section  now  is 

v^/A 
in  which  q'  is  found  from  the  formula 


This  value  of  q'  is  an  approximation  to  the  actual  dis- 
charge; from  it  v  is  found,  and  then  h,  after  which  the 
discharge  q  can  be  computed.  If  thought  necessary,  h 
may  be  recomputed  by  using  q  instead  of  q'  ';  but  this 
will  rarely  be  necessary. 

For  example,  the  small  weir  with  end  contractions 
used  in  the  hydraulic  laboratory  of  Lehigh  University 
prior  to  1896  had  B  =  7.  8  2  feet  and  £  =  2.5  feet.  The 
length  of  the  weir  b  was  adjustable  according  to  the  quan- 
tity of  water  delivered  by  the  stream.  On  April  10,  1888, 
the  value  of  b  was  1.330  feet,  and  values  of  H  ranged 
from  0.429  to  0.388  feet.  It  is  required  to  find  the  velocity 
v  and  the  head  h,  when  #  =  0.429  feet.  Here  the  co- 
efficient c  is  0.602  (Table  23),  hence  the  approximate  dis- 
charge per  second  is 

(f  =  0.602  Xf  X8.  02  X  1.33  Xo.429§ 
or  </  =  1.203  cubic  feet  per  second. 

The  mean  velocity  of  approach  then  is 
1.203 

""(2.5+0.4)7.82  =0-°53  feet  per  second> 

and  the  head  h  producing  this  velocity  is 

h  =  o.  01555  Xo.o532  =  0.00004  feet, 


148  FLOW  OVER  WEIRS  CHAP,  vi 

which  is  too  small  to  be  regarded,   since  the  hook  gage 
used  determined  the  heads  only  to  thousandths  of  a  foot. 

The  head  h  may  be   directly   expressed  in  terms   of 
the  discharge  by  substituting  for  v  its  value  q/A  ;  thus 

(62) 


and,  in  general,  this  expression  will  be  found  the  most 
convenient  one  for  computing  the  value  of  the  head  cor- 
responding to  the  velocity  of  approach. 

With  a  weir  opening  of  given  size  under  a  given  head 
H,  the  velocity  of  approach  is  less  the  greater  the  area 
of  the  section  of  the  feeding  canal,  and  it  is  desirable  in 
building  a  weir  to  make  this  area  large  so  that  the  velocity 
v  may  be  small.  For  large  weirs,  and  particularly  for 
those  without  end  contractions,  v  is  sometimes  as  large 
as  one  foot  per  second,  giving  ^=0.0155  feet,  and  these 
should  be  regarded  as  the  highest  values  allowable  if 
precision  of  measurement  is  required. 

Prob.  62.  Fteley  and  Stearns'  large  suppressed  weir  had  the 
following  dimensions:  6  =  ^  =  18.996  feet,  £  =  6.55  feet,  and  the 
greatest  measured  head  was  1.6038  feet.  Taking  ^  =  0.622,, 
compute  the  velocity  of  approach  and  its  velocity-head. 

ART.  63.     WEIRS  WITH  END  CONTRACTIONS 

Let  b  be  the  breadth  of  the  notch  or  length  of  the 
weir,  H  the  head  above  the  crest  measured  by  the  hook 
gage,  and  c  an  experimental  coefficient.  Then  if  there 
be  no  velocity  of  approach  the  discharge  per  second  is 


(63)  t 

But  if  the  mean  velocity  of  approach  at  the  section  where 
the  hook  is  placed  be  v,  let  h  be  the  head  which  would 
produce  this  velocity  as  computed  by  (62).  Then  the 
discharge  is 

)*  (63). 


ART.  63  WEIRS    WITH    END    CONTRACTIONS  149 

The  quantity  // 4-1.4/2  is  called  the  effective  head  on 
the  crest,  and,  as  shown  in  the  last  article,  h  is  usually 
small  compared  with  the  head  H. 

Table  23  contains  values  of  the  coefficient  of  discharge 
c  as  deduced  by  Hamilton  Smith,  from  a  discussion  of  the 
experiments  made  by  Lesbros,  Francis,  Fteley  and  Stearns, 
and  others.*  In  these  experiments  q  was  determined  by 
actual  measurement  in  a  tank  of  large  size,  and  the  other 
quantities  being  observed  the  coefficient  c  was  computed. 
Values  of  c  for  different  lengths  of  weir  and  for  different 
heads  were  thus  obtained,  and  after  plotting  them  mean 
curves  were  drawn  from  which  intermediate  values  were 
taken.  The  heads  in  the  first  column  are  the  effective 
heads  H+  i  .4/2;  but  as  h  is  small,  little  error  can  result  in 
using  H  as  the  argument  with  which  to  enter  the  table 
in  selecting  a  coefficient. 

It  is  seen  from  the  table  that  the  coefficient  c  increases 
with  the  length  of  the  weir,  which  is  due  to  the  fact  that 
the  end  contractions  are  independent  of  the  length.  The 
coefficient  also  increases  as  the  head  on  the  crest  diminishes. 
The  table  also  shows  that  the  greatest  variation  in  the 
coefficients  occurs  under  small  heads,  which  are  hence 
to  be  avoided  in  order  to  secure  accurate  measurements 
of  discharge. 

Interpolation  may  be  made  in  this  table  for  heads 
and  lengths  of  weirs  intermediate  between  the  values 
given,  regarding  the  coefficient  to  vary  uniformly  be- 
tween the  values  given.  When  coefficients  are  frequently 
required  for  a  weir  of  given  length  it  will  be  best  to  make 
out  a  special  table  for  that  weir  and  to  diagram  the  re- 
sults to  a  large  scale  on  cross-section  paper,  so  that  inter- 
polation for  different  heads  can  be  more  readily  made. 

As  an  example  of  the  use  of  the  formulas  and  Table 
23,  let  it  be  required  to  find  the  discharge  per  second 

*  Hydraulics  (London  and  New  York,  1884),  p.   132. 


150  FLOW  OVER  WEIRS  CHAP,  vi 


over  a  weir  4  feet  long  when  the  head  H  is  0.457 
there  being  no  velocity  of  approach.  From  the  table 
the  coefficient  of  discharge  is  0.614  for  77  =  0.4  and  0.6095 
for  77  =  o.5,  which  gives  about  0.612  when  77  =  0.457. 
Then  the  discharge  per  second  is 

q  =  0.612X1X8.02X4X0.457!  =4.04  cubic  feet. 

If  the  width  of  the  feeding  canal  be  7  feet,  and  its  depth 
below  the  crest  be  1.5  feet,  the  velocity-head  is 

0.00134  feet. 


The  effective  head  now  becomes  77+1.4^  =  0.459  feet, 
and  the  discharge  per  second  over  the  weir  is 

q  =0.612  XfX8.  02  X4X  0.459!  =4.07  cubic  feet. 

It  is  to  be  observed  that  the  reliability  of  these  computed 
•discharges  depends  upon  the  precision  of  the  observed 
quantities  and  upon  the  coefficient  c\  this  is  probably 
liable  to  an  error  of  one  or  two  units  in  the  third  decimal 
place,  which  is  equivalent  to  a  probable  error  of  about 
three-tenths  of  one  percent.  On  the  whole,  regarding 
the  inaccuracies  of  observation,  a  probable  error  of  one 
percent  should  at  least  be  inferred,  so  that  the  value 
9  =  4.07  cubic  feet  per  second  should  strictly  be  written 
9  =  4.07  ±0.04,  that  is,  the  discharge  per  second  has  4.07 
cubic  feet  for  its  most  probable  value,  and  it  is  as  likely 
to  be  between  the  values  4.03  and  4.11  as  to  be  outside 
of  those  limits. 

In  very  precise  work  the  value  of  the  acceleration  g 
should  be  computed  from  formula  (7)!  for  the  particular 
latitude  and  elevation  above  sea  level  where  the  weir  is 
located. 

Prob.  63a.  A  weir  in  north  latitude  40°  24'  and  395  feet 
above  sea  level  has  a  length  of  2.5  feet  .  Compute  the  dis- 
charges over  it,  the  feeding  canal  having  the  width  6  feet  and  the 


ART.  64  WEIRS   WITHOUT   END    CONTRACTIONS  151 

depth  below  crest   1.6  feet,  when  the  heads  on  the  crest  are 
0.314,  0.315,  and  0.316  feet. 

Prob.  636.  Compute  the  coefficient  of  discharge  for  the  fol- 
lowing experiment  by  Francis,  in  which  q  was  found  by  actual 
measurement  in  a  large  tank:  6  =  9.997  feet,  -8  =  13.96  feet, 
G  =  4. 19  feet,  H=  1.5243  feet,  2g  =  64.3236,  and  3  =  61.282 
cubic  feet  per  second. 


ART.  64.     WEIRS  WITHOUT  END  CONTRACTIONS 

For  weirs  without  end  contractions,  or  suppressed 
weirs  as  they  are  often  called,  when  there  is  no  velocity 
of  approach,  the  discharge  per  second  is 


and  when  there  is  velocity  of  approach, 

(64) 


Here  the  notation  is  the  same  as  in  the  last  article,  and 
c  is  to  be  taken  from  Table  25,  which  gives  the  coefficients 
of  discharge  as  deduced  by  Smith,  in  1888. 

It  is  seen  that  the  coefficients  for  suppressed  weirs 
are  greater  than  for  those  with  end  contractions:  this 
of  course  should  be  the  case,  as  contractions  diminish 
the  discharge.  They  decrease  with  the  length  of  the 
weir,  while  those  for  contracted  weirs  increase  with  the 
length.  Their  greatest  variation  occurs  under  low  heads, 
where  they  rapidly  increase  as  the  head  diminishes.  It 
should  be  observed  that  these  coefficients  are  not  reliable 
for  lengths  of  weirs  under  4  feet,  owing  to  the  few  ex- 
periments which  have  been  made  for  short  suppressed 
weirs.  Hence,  for  small  quantities  of  water,  weirs  with 
end  contractions  should  be  built  in  preference  to  sup- 
pressed weirs.  For  a  weir  of  infinite  length  it  would  be 
immaterial  whether  end  contractions  exist  or  not;  hence 
for  such  a  case  the  coefficients  lie  between  the  values 


152  FLOW  OVER  WEIRS  CHAP,  vr 

for  the  ip-foot  weir  in  Table  23  and  those  for  the  1 9-foot 
weir  in  Table  25. 

For  a  numerical  illustration  a  suppressed  weir  having 
the  same  dimensions  as  in  the  example  of  the  last  article 
will  be  used,  namely,  6  =  4  feet,  G  =  i.$  feet,  and  H  =  0.457 
feet.  The  coefficient  is  found  from  Table  25  to  be  0.630; 
then  for  no  velocity  of  approach  the  discharge  per  sec- 
ond is 

<?  =  0.630X1  X8.02  X 4X0. 45 7!  =  4.16  cubic  feet. 

Here  the  width  B  is  also  4  feet;    the  head  corresponding 
to  the  velocity  of  approach  then  is  by  (62), 

=°-°°44  feet, 


and  the  effective  head  on  the  crest  is 

//+ ij/z  =0.463  feet, 
from  which  the  discharge  per  second  is 

q  =  o. 630X1X8. 02  X4Xo. 463!  =4.24  cubic  feet. 

This  shows  that  the  velocity  of  approach  exerts  a  greater 
influence  upon  the  discharge  than  in  the  case  of  a  weir 
with  end  contractions. 

Prob.  64.  Compute  the  discharge  per  second  over  a  weir 
without  end  contractions  when  6  =  9.995  ^eet>  ^==o-7955  feet,. 
£  =  4.6  feet. 

ART.  65.     FRANCIS'  FORMULAS 

The  formulas  most  extensively  used  for  computing 
the  flow  through  weirs  are  those  established  by  Francis 
in  1854*  from  the  discussion  of  his  numerous  and  carefully 
conducted  experiments,  but  as  they  are  stated  without 
tabular  coefficients  they  are  to  be  regarded  as  giving  only 
mean  approximate  results.  The  experiments  were  made 
on  large  weirs,  most  of  them  10  feet  long,  and  with  heads, 

*  Lowell  Hydraulic  Experiments  (4th  edition,  New  York,  1883),  p.  133. 


ART.  65  FRANCIS  '    FORMULAS  153 

ranging  from  0.4  to  1.6  feet,  so  that  the  formulas  apply 
particularly  to  such,  rather  than  to  short  weirs  and  low 
heads.  The  length  b  and  the  head  H  being  expressed  in 
feet,  the  discharge  per  second,  when  there  is  no  velocity 
of  approach,  is,  for  weirs  without  end  contractions,  or 
suppressed  weirs, 

?  =  3.33fcH*  (65), 

and  for  weirs  with  end  contractions, 

q  =  3.33(b-o.2H)H*  (65)2 

Here  it  is  regarded  that  the  effect  of  each  end  contraction 
is  to  diminish  the  effective  length  of  the  weir  by  o.iH. 
In  these  formulas  b  and  H  must  be  taken  in  feet  and  q 
will  be  in  cubic  feet  per  second. 

Francis'  method  of  correcting  for  velocity  of  approach 
differs  from  that  of  Smith,  and  is  the  same  as  that  ex- 
plained in  Art.  .51.  The  head  h  causing  the  velocity  of 
approach  is  computed  in  the  usual  way,  and  then  the 
formulas  are  written,  for  weirs  without  end  contractions, 


and  for  weirs  with  end  contractions, 


It  is  necessary  that  this  method  of  introducing  the  velocity 
of  approach  should  be  strictly  observed,  since  the  mean 
number  3.33  was  deduced  for  this  form  of  expression. 

It  is  seen  that  the  number  3.33  is  c.f\/2g,  where  c 
is  the  true  coefficient  of  discharge.  The  88  experiments 
from  which  this  mean  value  was  deduced  show  that  the 
coefficient  3.33  actually  ranged  from  3.30  to  3.36,  so  that 
by  the  use  of  the  mean  value  an  error  of  one  percent  in 
the  computed  discharge  may  occur.  When  such  an  error 
is  of  no  importance  the  formula  may  be  safely  used  for 
weirs  longer  than  4  feet  and  heads  greater  than  0.4  feet. 


154  FLOW  OVER  WEIRS  CHAP,  vi 

Prob.  65.  Find  by  Francis'  formulas  the  discharge  when 
£  =  7  feet,  b  =  4  feet,  #  =  0.457  &&>,  and  G=i.$  feet,  the  weir 
being  one  with  end  contractions. 

ART.  66.     SUBMERGED  WEIRS 

When  the  water  on  the  down -stream  side  of  the  weir 
is  allowed  to  rise  higher  than  the  level  of  the  crest  the 
weir  is  said  to  be  submerged.  In  such  cases  an  entire 
change  of  condition  results,  and  the  preceding  formulas 
are  inapplicable.  Let  H  be  the  head  above  the  crest 
measured  up-stream  from  the  weir  by  the  hook  gage  in 
the  usual  manner,  and  let  H'  be  the  head  above  the  crest 
of  the  water  down-stream  from  the  weir  measured  by 
a  second  hook  gage.  If  H  be  constant,  the  discharge 

is    uninfluenced    until    the    lower 
water    rises    to    the    level    of    the 
crest,    provided    that    free    access 
|T  of    air    is    allowed    beneath    the 

descending   sheet   of   water.     But 
as  soon  as  it  rises  slightly  above 

the  crest  so  that  H'  has  small  values,  the  contraction  is 
suppressed  and  the  discharge  hence  increased.  As  Hr 
increases,  however,  the  discharge  diminishes  until  it  be- 
comes zero  when  H'  equals  H.  Submerged  weirs  cannot 
be  relied  upon  to  give  precise  measurements  of  discharge 
on  account  of  the  lack  of  experimental  knowledge  regard- 
ing them,  and  should  hence  always  be  avoided  if  possible. 

The  following  method  for  estimating  the  discharge 
over  submerged  weirs  without  end  contractions  is  taken 
from  the  discussion  given  by  Herschel*  of  the  experiments 
made  by  Francis  and  by  Fteley  and  Stearns.  The  ob- 
served head  H  is  first  multiplied  by  a  number  n,  which 
depends  upon  the  ratio  of  H'  to  H,  and  then  the  discharge 

*  Transactions  of  the  American  Society  of  Civil  Engineers,    1885,    vol. 
14,  p.  194. 


ART.  66  SUBMERGED    WEIRS  155 

is  to  be  computed  by  using  the  modified  Francis*  formula 


The  values  of  n  deduced  by  Herschel  are  given  in  Table 
27.  They  are  liable  to  a  probable  error  of  about  one 
unit  in  the  second  decimal  place  when  Hf  is  less  than 
0.2  H,  and  to  greater  errors  in  the  remainder  of  the  table, 
values  of  n  less  than  0.70  being  in  particular  uncertain.  It 
is  seen  that  H'  may  be  nearly  one-fifth  of  H  without  affect- 
ing the  discharge  more  than  two  percent. 

A  rational  formula  for  the  discharge  over  submerged 
weirs  may  be  deduced  in  the  following  manner.  .  The 
theoretic  discharge  may  be  regarded  as  composed  of  two 
portions,  one  through  the  upper  part  H  —  H',  and  the 
other  through  the  lowrer  part  H'.  The  portion  through 
the  upper  part  is  given  by  the  usual  weir  formula,  H—H' 
being  the  head,  or 


and  that  through  the  lower  part  is  givfen  by  the  formula 
for  a  submerged  orifice  (Art.  52),  in  which  b  is  the  breadth, 
H'  the  height,  and  H  —  H'  the  effective  head,  or 


The  addition  of  these  gives  the  total  theoretic  discharge, 

Q  =  %V^b(H-H'^  +  V^bH'(H-H')* 
which  may  be  put  into  the  more  convenient  form, 


ti  r  £ 

The  actual  discharge  per  second  may  now  be  written, 


in  which  c  is  the  coefficient  of  discharge. 

Fteley  and  Stearns  adopt  the  above  formula  for  the 
discharge,  or  placing  M  for  c  .\\/  2°,  they  write,* 

q  =  Mb(H  +  i2H')(H-H')l  (66)2 

*  Transactions  American  Society  Civil  Engineers,  1883,  vol.  12,  p.  103. 


156 


FLOW  OVER  WEIRS 


CHAP.  VI 


and  from  their  experiments  deduce  the  following  values 

of  the  coefficient  M  : 

for     H'/H  =  o.oo     0.04     0.08     0.12     0.16     0.2       0.3 

M=3-33     3-35     3-37     3-35     3-32     3^8     3.21 
for     H'/H  =  o.4       0.5       0.6       0.7       0.8       0.9       i.o 

M  =  3-I5     3-11     3-°9     3-°9     3-J2     3-J9     3-33 

These  are  for  suppressed  weirs;    for  contracted  weirs  few 
or  no  experiments  are  on  record. 

In  what  has  thus  far  been  said  velocity  of  approach 
has  not  been  considered.  This  may  be  taken  into  account 
in  the  usual  way  by  determining  the  velocity -head  h, 
and  thus  correcting  H.  But  it  is  unnecessary,  on  account 
of  the  limited  use  of  submerged  weirs,  and  the  consequent 
lack  of  experimental  data,  to  develop  this  branch  of  the 
subject.  What  has  been  given  above  will  enable  a  prob- 
able estimate  to  be  made  of  the  discharge  in  cases  where 
the  water  accidentally  rises  above  the  crest,  and  further 
than  this  the  use  of  submerged  weirs  cannot  be  recom- 
mended. 

Prob.  66.  Compute  by  two  methods  the  discharge  over  a 
submerged  weir  when  6  =  8,  #  =  0.46,  and  Hf  =  o.22  feet. 


ART.  67.     ROUNDED  AND  WIDE  CRESTS 

When  the  inner  edge  of  the  crest  of  a  weir  is  rounded, 
as  at  a  in  Fig.  67,  the  discharge  is  materially  increased 
as  in  the  case  of  orifices  (Art.  54),  or  rather  the  coefficients 

of  discharge  become  much 
larger  than  those  given  for 
the  standard  sharp  crests. 
The  degree  of  rounding 


influences     so     much     the 
FIG.  67 

amount  of  increase  that  no 

definite  values  can  be  stated,  and  the  subject  is  here  merely 
mentioned  in  order  to  emphasize  the  fact  that  a  rounded 


ART.  68  WASTE    WEIRS    AND    DAMS  157 

inner  edge  is  always  a  source  of  error.  If  the  radius  of 
the  rounded  edge  is  small,  the  sheet  of  escaping  water 
is  at  a  point  below  the  top  (a  in  the  figure),  which  has  the 
practical  effect  of  increasing  the  measured  head  by  a 
constant  quantity.  The  experiments  of  Fteley  and  Stearns 
show  that  when  the  radius  is  less  than  one-half  an  inch, 
the  discharge  can  be  computed  from  the  usual  weir 
formula,  seven-tenths  of  the  radius-  being  first  added  to 
the  measured  head  H. 

Two  wide-crested  weirs  with  square  inner  corners  are 
shown  in  Fig.  67,  the  one  at  B  being  of  sufficient  width 
so  that  the  descending  sheet  may  just  touch  the  outer 
edge,  causing  the  flow  to  be  more  or  less  disturbed,  while 
that  at  C  has  the  sheet  adhering  to  the  crest  for  some 
distance.  In  both  cases  the  crest  contraction  occurs, 
although  water  instead  of  air  may  fill  the  space  above 
the  inner  corner.  For  B  the  discharge  may  be  equal 
to  or  greater  than  that  of  the  standard  weir  having  the 
same  head  //,  depending  upon  whether  the  air  has  or  has 
not  free  access  beneath  the  sheet  in  the  space  above  the 
crest.  For  C  the  discharge  is  always  less  than  that  of 
the  standard  weir  with  sharp  crest. 

Table  28  is  an  abstract  from  the  results  obtained  by 
Fteley  and  Stearns,*  and  gives  the  corrections  in  feet  to 
be  subtracted  from  the  depths  on  a  wide  crest,  like  C 
in  Fig.  67,  in  order  to  obtain  the  depths  on  a  standard 
sharp-crested  suppressed  weir  giving  the  same  discharge. 

Prob.  67.  Compute"  the  discharge  ov~er  a  crest  1.5  feet  wide 
for  a  weir  10  feet  long  when  the  head  is  0.850  feet. 

ART.  68.     WASTE  WEIRS  AND  DAMS 

Waste  weirs  are  constructed  at  the  sides  of  reservoirs 
in  order  to  allow  the  surplus  water  to  escape.  They  are 
usually  arranged  so  that  the  end  contractions  are  suppressed. 

*  Transactions  American  Society  Civil  Engineers,  1883,  vol.  12,  p.  96. 


158  FLOW  OVER  WEIRS  CHAP,  vi 

When  the  crest  is  narrow  and  the  front  vertical,  so  that 
the  descending  sheet  of  water  has  air  upon  its  lower  side, 
the  discharge  is  approximately  given  by  Francis'  weir 
formula  (Art.  65), 


in  which  b  is  the  length  of  the  crest,  and  H  the  head 
measured  some  distance  back  from  the  crest.  When  the 
crest  is  wide  and  the  approach  to  it  is  inclined,  as  is  often 
the  case,  the  discharge  is  somewhat  smaller.  For  a  crest 
about  three  feet  wide  and  level,  with  an  inclined  approach 
back  of  it,  Francis  deduced  from  his  experiments, 


which,  for  a  head  of  one  foot,  gives  a  discharge  ten  percent 
less  than  that  of  the  first  formula. 

In  constructing  a  waste  weir  the  discharge  q  is  generally 
known  or  assumed,  and  it  is  required  to  determine  b  and 
H.  The  latter  being  taken  at  i,  2,  or  3  feet,  as  may  be 
judged  safe  and  proper,  b  is  found  by  one  of  these  formulas. 
For  example,  if  the  crest  be  wide,  q  be  87  cubic  feet  per 
second,  and  Hbe  two  feet,  then 

log  b  =log  87  -log  3.01  -  1.53  log  2 

from  which  log  b  =  1.0004,  whence  6  =  10.0  feet.  If,  how- 
ever, the  crest  be  narrow,  the  first  formula  gives  6  =  9.2 
feet.  Evidently  no  great  precision  is  needed  in  comput- 
ing the  length  of  a  waste  weir,  since  it  is  difficult  to  de- 
termine the  exact  discharge  which  is  to  pass  over  it,  and 
an  ample  factor  of  safety  should  be  introduced  to  cover 
unusual  floods. 

The  above  formulas  may  be  used  for  obtaining  the 
approximate  flow  of  a  stream  in  which  a  dam  with  level 
crest  has  been  built.  The  water,  however,  is  often  re- 
ceived upon  an  apron  of  timber  or  masonry,  and  the 
inclination  of  this,  as  well  as  the  inclination  of  the  ap- 


ART.  68  WASTE    WEIRS   AND   DAMS  159 

proach  to  the  crest,  materially  modifies  the  discharge. 
The  formula 

q  =  c.$\/2g  &H*-w6H*  (68) t 

is  usually  employed  for  dams  and  it  is  found  that  the 
value  of  M,  for  English  measures,  may  range  under  differ- 
ent circumstances  from  2.5  to  4.2.  This  is  modified  below 
for  velocity  of  approach  (Art.  62). 

Experiments  were  made  by  Bazin  in  1897  *  on  dams 
from  1.6  to  2.5  feet  high  with  heads  of  water  on  the  crests 
ranging  from  0.2  to  1.4  feet.  For  the  case  of  Fig.  68a 


FIG.  68a  FIG.  686  FIG.  68c 

the  approach  had  an  inclination  of  i  on  2  and  the  front 
was  vertical;  when  the  width  of  the  crest  was  0.33  feet, 
the  coefficient  M  varied  from  3.24  to  4.12  as  the  head 
increased  from  0.27  to  1.41  feet;  when  the  width  of  the 
crest  was  0.66  feet,  M  varied  from  3.10  to  3.89  for  similar 
heads.  For  the  case  of  Fig.  686  both  approach  and  apron 
had  slopes  of  i  on  2  and  the  crest  was  0.66  feet  wide; 
here  M  increased  from  2.83  to  3.75  as  the  head  ranged 
from  0.22  to  1.42  feet.  For  Fig.  68c,  with  a  crest  2.62 
feet  wide,  M  ranged  from  2.47  to  2.76,  but  when  the  up- 
stream corner  was  rounded  to  a  radius  of  4  inches  it 
ranged  from  2.71  to  3.12.  Here  it  is  seen  that  widening 
the  crest  decreases  the  discharge,  as  already  noted  in  Art. 
67,  and  that  the  apron  produces  a  similar  influence. 

Experiments  on  a  larger  scale  were  made  by  Rafter, 
in  1898,  for  the  U.  S.  Deep  Waterways  Commission 
at  the  canal  of  the  Cornell  hydraulic  laboratory,  in  which 
the  flow  over  dams  was  measured  by  a  standard  weir. 

*  Annales  des  ponts  et  chaussees,  1 898 ;    translated  by  Rafter  in  Trans- 
actions American  Society  Civil  Engineers,  1900,  vol.  44,  p.  254. 


160  FLOW  OVER  WEIRS  CHAP,  vi 

The  results  of  these  experiments  are  given  in  Table  29, 
the  first  five  being  for  darns  of  the  form  shown  in  Fig. 
68a,  the  next  three  for  dams  like  Fig.  686,  and  the  next 
four  for  dams  like  Fig.  68c,  those  marked  with  an  asterisk 
having  the  up-stream  corner  rounded  to  a  radius  of  4 
inches.  The  last  line  of  the  table  refers  to  a  section  whose 
top  was  five  feet  wide  and  rounded  to  a  radius  of  3.37 
feet,  the  rounding  beginning  on  the  up-stream  side  one 
foot  below  the  crest.  The  height  of  these  dams  varied 
from  4.56  to  4.91  feet,  and  the  length  of  the  crest  was  in 
all  cases  6.58  feet.* 

Rafter  also  made  experiments   on  some  other  forms 
of  dams.     The  one  shown  in  Fig.  QSd  had  a  vertical  front 


FIG.  QSd  FIG.  6&? 

4.57  feet  deep,  and  the  two  back  slopes  were  i  on  6  and 
i  on  |,  the  width  of  the  former  being  4.5  feet;  the  values 
of  M  for  this  case  ranged  from  3.33  to  3.46  for  heads  rang- 
ing from  i.o  to  6.0  feet.  The  one  shown  in  Fig.  QSe  had 
a  total  width  of  about  23  feet  and  a  height  of  4.53  feet, 
the  slopes  of  the  approach  and  apron  being  i  on  6,  and 
that  just  below  the  crest  about  i  on  J,  the  vertical  depth 
of  this  being  0.75  feet ;  for  this  the  mean  values  of  M  ranged 
from  3.07  to  3.27  for  heads  ranging  from  i.o  to  6.0  feet, 
the  smaller  coefficients  being  due  to  the  contact  of  the 
water  with  the  apron. 

By  the  use  of  these  coefficients  the  discharge  of  a 
small  stream  over  a  dam  under  medium  heads  may  be 
computed  with  a  degree  of  precision  probably  as  high 
as  other  methods  will  give.  After  finding  q  from  (68) j 
the  head  h  corresponding  to  the  velocity  of  approach  is 
to  be  determined  by  (62)  x  and  then 

q  =  ub(H  +  h)*  (68), 

*  Transactions  American  Society  Civil  Engineers,  1900,  vol.  44,  p.  266. 


ART.  69  THE    SURFACE    CURVE  161 

is  to  be  used  for  obtaining  a  more  precise  value  of  the 
discharge.  For  example,  suppose  a  darn  of  the  form  of 
Fig.  68a  to  be  226  feet  long  with  a  back  slope  of  i  on  4, 
and  to  have  a  head  of  1.25  feet  on  its  crest.  From  Table 
29  there  is  found  by  interpolation  M=3.45,  and  then  by 
(^8)t  the  approximate  discharge  is  1090  cubic  feet  per 
second.  Let  the  stream  be  150  feet  wide  and  6.5  feet 
-deep  at  the  place  where  the  head  is  measured,  then  from 
(62)  the  head  causing  the  velocity  of  approach  is  0.02 
feet,  and  from  (68) 2  the  discharge  is  mo  cubic  feet  per 
.second,  which  is  to  be  regarded  as  liable  to  a  probable 
error  of  five  percent. 

Prob.  68.  Find  the  length  of  a  waste  weir  which  will  be 
ample  to  discharge  a  rainfall  of  one  inch  per  hour  on  a  drainage 
area  of  3.65  square  miles,  the  head  on  the  crest  of  the  weir 
being  2.12  feet.  Also  when  the  head  is  4.24  feet. 


ART.  69.     THE  SURFACE  CURVE 

The  surface  of  the  water  above  a  weir  or  dam  assumes 
a  curve  whose  equation  is  a  complex  one,  but  some  of 
the  laws  that  govern  the  drop  in  the  plane  of  the  crest 
may  be  deduced.  Let  H  be  the  head  on  the  level  of  the 
crest  measured  in  perfectly  level  water  at  some  distance 
back  of  the  weir,  and  let  d  be  the 
depression  or  drop  of  the  curve  be- 
low this  level  in  the  plane  of  the 
weir  (Fig.  69).  Then  .the  discharge  per 
second  q  can  be  expressed  in  terms 
of  H  and  d  by  formula  (51)4,  placing 
H  for  h2  and  d  for  hlt  and  making  FlG-  69 

7z0  =  o.     This  formula  becomes,  after  replacing  \\Tzg  by  M, 
and  Q  by  q, 


This   expression,   it  may  be  remarked,   is  the  true  weir 


162  FLOW  OVER  WEIRS  CHAP,  vi 

formula,  and  only  the  practical  difficulties  of  measuring 
H  and  d  prevent  its  use.     This  may  be  written 


from  which  the  drop  d  in  the  plane  of  crest  of  the  weir 
can  be  found.  Let  B  be  the  breadth  of  the  feeding  canal, 
G  its  depth  below  the  crest,  and  v  the  mean  velocity  of 
approach;  then  also 

q  =  B(G+H)v 

and  inserting  this  in  the  expression  for  d%  it  becomes 

d*  =  PI%-JI-b(G+H)v  (69) 

which  is  an  expression  for  the  drop  of  the  curve  in  terms 
of  the  dimensions  of  the  weir,  the  total  head,  and  the  ve- 
locity of  approach. 

The  approximate  value  of  the  coefficient  M  is  about 
3.3  for  English  measures,  but  precise  values  of  d  cannot 
be  computed  unless  M  and  H  are  known  with  accuracy. 
The  formula,  however,  serves  to  exemplify  the  laws  which 
govern  the  drop  of  the  curve  in  the  plane  of  the  weir. 
It  shows  that  the  drop  increases  with  the  head  on  the 
crest  and  with  the  length  of  a  contracted  weir,  that  it  de- 
creases with  the  breadth  and  depth  of  the  feeding  canal, 
and  that  it  decreases  with  the  velocity  of  approach.  It 
also  shows  for  suppressed  weirs,  where  B  =b,  that  the 
drop  is  independent  of  the  length  of  the  wreir.  All  of 
these  laws  except  the  last  have  been  previously  deduced 
by  the  discussion  of  experiments. 

The  path  of  the  stream  after  leaving  the  weir  is  closely 
that  of  a  parabola.  In  the  plane  of  the  crest  the  mean 
velocity  is 

V  =  q/b(H-d) 

and  the  direction  of  this  may  be  taken  as  approximately 
horizontal.  The  range  of  a  stream  on  a  horizontal  plane 
•  at  the  distance  y  below  the  middle  of  the  weir  notch  is 


ART.  70  TRIANGULAR  WEIRS  163 

then  readily  found.  For,  if  x  be  this  range  which  is 
reached  in  the  time  t,  then  x  =  Vt,  and  also  y  =  %gt2,  whence, 
by  the  elimination  of  /,  there  results  gx*  =  2  V2y,  and 
accordingly  the  horizontal  range  at  the  depth  y  is 

H*-d*   \vy 

X  =  M-rj "TV 

H-d   v  g 

in  which  d  is  given  by  (69).  For  example,  take  a  case 
where  #  =  3  feet,  £  =  23  feet,  and  ^  =  0.5  feet  per  second. 
From  (69)  the  value  of  d  is  found  to  be  1.17  feet.  Now, 
if  y  =  $o  feet,  the  last  formula  gives  #  =  6.i  feet,  which 
is  the  distance  of  the  middle  of  the  stream  from  the  ver- 
tical plane  through  the  crest  at  50  feet  below  that  crest. 

Prob.  69.  In  the  above  example  what  velocity  of  approach 
is  necessary  in  order  that  there  may  be  no  drop  in  the  plane 
of  the  crest.  What  is  the  range  for  this  case  ? 


ART.  70.     TRIANGULAR  WEIRS 

Triangular  weirs  are  sometimes  used  for  the  measure- 
ment of  water,  the  arrangement  being  as  shown  in  Fig. 
70.  Let  b  be  the  width  of 
the  orifice  at  the  water  level, 
and  H  the  head  of  water  on 
the  vertex.  Let  an  elemen- 
tary strip  of  the  depth  dy  ""  FlG  7Q 

be  drawn  at  a  distance  y 
below  the  water  level.  From  similar  triangles  the  length 
of  this  strip  is  (H  —  y)b/H  and  the  elementary  discharge 
through  it  then  is 


The  integration  of  this  between  the  limits  H  and  o  gives 
the  theoretic  discharge  through  the  triangular  weir,  namely, 


164  FLOW  OVER  WEIRS  CHAP,  vi 

If  the  sides  of  the  triangle  are  equally  inclined  to  the 
vertical,  as  should  be  the  case  in  practice,  and  if  this 
angle  be  a,  the  surface  width  b  may  be  expressed  in  terms 
of  a  and  H  ,  so  that  the  last  formula  becomes 

Q-Atanor.v^g.//*  (70)2 

The  discharge  is  thus  equal  to  a  constant  multiplied  by 
the  2-J  power  of  the  measured  depth. 

Triangular  weirs  are  used  but  little,  as  in  general  they 
are  only  convenient  when  the  quantity  of  water  to  be 
measured  is  small.  Such  a  weir  must  have  sharp  inner 
corners,  so  that  the  stream  may  be  fully  contracted,  and 
the  sides  should  have  equal  slopes.  The  angle  at  the 
lower  vertex  should  be  a  right  angle,  as  this  is  the  only 
case  for  which  coefficients  are  known  with  precision. 
The  depth  of  water  above  this  lower  vertex  is  to  be  measured 
by  a  hook  gage  in  the  usual  manner  at  a  point  several 
feet  up-stream  from  the  notch.  Making  the  angle  at 
the  vertex  a  right  angle,  and  applying  a  coefficient,  the 
actual  discharge  per  second  is  given  by  the  expression 


in  which  H  is  the  head  of  water  above  the  vertex.  Ex- 
periments made  by  Thomson  *  indicate  that  the  coefficient 
c  varies  less  with  the  head  than  for  ordinary  weirs;  this, 
in  fact,  was  anticipated,  since  the  sections  of  the  stream 
are  similar  in  a  triangular  notch  for  all  values  of  H,  and 
hence  the  influence  of  the  contractions  in  -diminishing 
the  discharge  should  be  approximately  the  same.  As 
the  result  of  his  experiments  the  mean  value  of  c  for  heads 
between  0.2  and  0.8  feet  may  be  taken  as  0.592,  and  hence 
the  mean  discharge  in  cubic  feet  per  second  through  a 
right-angled  triangular  weir  may  be  written 


*  British  Association  Report,  1858,  p.  133. 


ART.  71  TRAPEZOIDAL  WEIRS  165 

in  which,  as  usual,  H  must  be  expressed  in  feet.  About 
4  feet  is  probably  the  greatest  practicable  value  for  H, 
and  this  gives  a  discharge  of  only  81.0  cubic  feet  per 
second.  If  velocity  of  approach  exists,  H  in  this  formula 
should  be  replaced  by  H+i.tft,  as  for  rectangular  weirs 
with  end  contractions. 

Prob.  70.  A  triangular  orifice  in  the  side  of  a  vessel  has  a 
horizontal  base  b  and  an  altitude  d,  the  head  of  water  on  the 
base  being  h  and  that  on  the  vertex  being  h-\-d.  Show  that  the 
theoretic  discharge  is  T1-3-V/Jg(6/J)[4(/i+^)^-(4/j+  iod)h$]. 


ART.  71.     TRAPEZOIDAL  WEIRS 

Trapezoidal  weirs  are  sometimes  used  instead  of  rect- 
angular ones,  as  the  coefficients  vary  less  in  value.  The 
theoretic  discharge  through  a  trapezoidal  weir  which  has 
the  length  b  on  the  crest,  the 
head  H,  and  the  length  b  +  2Z 
on  the  water  surface,  as  seen 
in  Fig.  71,  is  the  sum  of  the 
discharges  through  a  rect- 
angle of  area  bH  and  a 
triangle  of  area  zH.  Taking  the  former  from  (61)!  and 
the  latter  from  (70)  2,  and  replacing  tan  a  by  z/H 


is  the  theoretic  discharge.  Here  z/H,  which  is  the  slope 
of  the  ends,  may  be  any  convenient  number,  and  it  is 
usually  taken  as  i,  as  first  recommended  by  Cippoletti.* 

The  reasoning  from  which  this  conclusion  was  derived 
is  based  upon  Francis'  rule  that  the  two  end  contractions 
in  a  standard  rectangular  weir  diminish  the  discharge 
by  a  mean  amount  3.33X0.2/7*  (Art.  65),  or  in  general 
by  the  amount  £.f\/2gXo.2/-A  If  the  sides  be  sloped, 

*  Cippoletti,  Canal  Villoresi,  1887  ;  see  Engineering  Record,  Aug.  13,  1892. 


166  FLOW  OVER  WEIRS  CHAP,  vi 

however,  the  discharge  through  the  two  end  triangles  is 
c.-f^\/2gX!2H^.  If,  now,  the  slope  is  just  sufficient  so 
that  the  extra  discharge  balances  the  effect  of  the  end 
contractions,  these  two  quantities  are  equal.  Equating 
them,  and  supposing  that  c  has  the  same  value  in  each, 
there  results  z  =  \H.  Hence  for  such  a  trapezoidal  weir 
the  discharge  should  be  the  same  as  that  from  a  sup- 
pressed rectangular  weir  of  length  fr,  or,  according  to 
Francis,  9  =  3.  33^*.  Cippoletti,  however,  concluded  from 
his  experiments  that  the  coefficient  should  be  increased 
about  one  percent,  and  he  recommended 

5=3.36767/1 

as  the  formula  for  discharge  over  such  a  trapezoidal  weir 
when  no  velocity  of  approach  exists. 

Experiments  by  Flinn  and  Dyer*  indicate  that  the 
coefficient  3.367  is  probably  a  little  too  large.  In  32 
tests  with  trapezoidal  weirs  of  from  3  to  9  feet  length 
on  the  crest  and  under  heads  ranging  from  0.2  to  1.4  feet, 
they  found  28  to  give  discharges  less  than  the  formula, 
the  percentage  of  error  being  over  3  percent  in  eight  cases. 
The  four  cases  in,  which  the  discharge  was  greater  than 
that  given  by  the  formula  show  a  mean  excess  of  about 
3.5  percent.  The  mean  deficiency  in  all  the  32  cases  was 
nearly  2  percent.  These  experiments  are  not  very  precise, 
since  the  actual  discharge  was  computed  by  measure- 
ments on  a  rectangular  weir,  so  that  the  results  are  neces- 
sarily affected  by  the  errors  of  two  sets  of  measurements. 
Cippoletti  's  formula,  given  above,  may  hence  be  allowed 
to  stand  as  a  fair  one  for  general  use  with  trapezoidal 
weirs  in  which  the  slope  of  the  ends  is  J.  It  can,  of  course, 
be  written  in  the  form 


where  the  coefficient  c  has  the  mean  value  0.629,  while 

*  Transactions  American  Society  of  Civil  Engineers,  1  894,  vol.  32,  pp.  9-33. 


ART.  72  COMPUTATIONS    IN    THE    METRIC    SYSTEM       .  167 

Table  25  may  be  used  to  obtain  more  reliable  values  for 
special  cases. 

If  velocity  of  approach  exists,  H  in  this  formula  is  to 
be  replaced  by  H+i.^h  where  h  is  the  head  due  to  that 
velocity.  In  order  to  do  good  work,  however,  h  should 
not  exceed  0.004  feet.  Other  precautions  to  be  observed 
are  that  the  cross-section  of  the  canal  should  be  at  least 
seven  times  that  of  the  water  in  the  plane  of  the  crest, 
and  that  the  error  in  the  measured  head  should  not  be 
greater  than  one-third  of  one  percent.  On  the  whole, 
however,  the  coefficients  for  the  standard  rectangular 
weir  with  end  contractions  are  so  definitely  established, 
and  those  for  trapezoidal  weirs  so  imperfectly  known, 
that  the  use  of  the  latter  cannot  be  recommended  in  any 
case  where  the  greatest  degree  of  precision  is  required. 

The  above  formula  for  the  theoretic  discharge  may 
be  applied  to  the  Cippoletti  trapezoidal  weir  by  putting 
z  —  \H,  and  introducing  a  coefficient  ;  thus, 


is  a  formula  for  the  actual  discharge,  in  which  the  values 
of  c  are  probably  not  far  from  those  given  in  Table  23 
for  rectangular  contracted  weirs.  Here  the  term  o.2H/b 
shows  the  effect  of  the  two  end  triangles  in  increasing 
the  discharge. 

Prob.  71.  For  a  head  of  0.7862  feet  on  a  Cippoletti  weir  of 
4  feet  length  the  actual  discharge  in  420  seconds  was  3912.3 
cubic  feet.  Compute  the  discharge  by  the  formula  and  find 
the  percentage  of  error. 

ART.  72.     COMPUTATIONS  IN  THE  METRIC  SYSTEM 

The  formulas  for  discharge  in  Arts.  61-64  are  rational 
and  may  be  used  in  all  systems,  the  coefficients  c  being 
abstract  numbers.  In  the  metric  system  b  and  H  are 
often  expressed  in  centimeters  but  they  should  be  reduced 


168  FLOW  OVER  WEIRS  CHAP,  vr 

to  meters  for  use  in  the  formulas,  and  then  q  will  be  in 
cubic  meters  per  second.  The  mean  value  of  V  2g  is 
4.427  and  that  of  i/2g  is  0.05102. 

(Art.  62)  The  head  h  in  meters  corresponding  to  the 
mean  velocity  of  approach  is  to  be  computed  from  the 
formula 

h  =  o.o$io2(q/A)*  (72)  t 

in  which  A  is  in  square  meters.  For  example,  take  a 
weir  where  B  =  2oo,  £  =  90,  6  =  45.1,  H  =  26.2&  centimeters, 
and  0  =  0.620.  Then  by  (63)  t  the  discharge  q'  is  0.1112 
cubic  meters  per  second,  and  from  (72)  l  the  head  h  is 
0.0002  meters. 

(Art.  63)  Table  24  gives  values  of  the  coefficient  c 
for  weirs  with  end  contractions,  with  arguments  in  the 
metric  system.  Thus,  if  H  =  5.45  centimeters  and  6  =  0.45 
meters,  there  is  found,  by  interpolation,  c  =  0.620,  which 
is  liable  to  a  probable  error  of  about  two  units  in  the  third 
decimal  place. 

{Art.  64)  Coefficients  for  weirs  without  end  contrac- 
tions, with  metric  arguments,  are  given  in  Table  26, 
which  has  been  prepared  by  the  help  of  Table  25. 

(Art.  65)  When  b  and  H  are  in  meters  and  q  in  cubic 
meters  per  second,  Francis'  formula  for  suppressed  weirs- 
takes  the  form 

g  =  i.84Wf*  (72)* 

and  for  weirs  with  end  contractions, 


(72),. 
the  number  1.84  being  a  mean  value  of  c. 


(Art.  66)  Table  27  applies  to  any  system  of  measures, 
and  the  formula  g—  -i.84&(nH)*  then  gives  the  discharge 
in  cubic  meters  per  second,  if  6  and  H  be  in  meters.  The 
metric  values  of  m  for  use  in  (66)  are  found  by  multiplying. 
those  in  the  text  by  0.5522. 


ART.  72  METRIC    COMPUTATIONS  169 

(Art.  68)  The  formulas  of  the  first  paragraph  are 
transformed  into  metric  measures  by  replacing  3.33  by 
1.84  and  3.01  by  1.72.  For  formula  (68)1  the  value  of 
M  for  dams,  may  range  from  about  1.4  to  2.3.  Table  30 
gives  metric  values  of  M  as  deduced  from  the  experiments 
made  by  Bazin  in  1897,  and  by  Rafter  in  1898.  The 
explanation  of  this  table  is  in  all  respects  like  that  of 
Table  29.  All  values  of  M  given  in  Art.  68  may  be  re- 
duced to  metric  measures  by  multiplying  by  0.5522,  this 
being  the  ratio  of  the  value  of  V '  2g  expressed  in  meters 
to  that  expressed  in  feet. 

(Art.  70)  The  metric  formula  for  discharge  over  the 
triangular  weir  is  q  =  1.40^. 

(Art.  71)  The  metric  formula  for  Cippoletti's  trape- 
zoidal weir  takes  the  form  q  =  i.S6bH%. 

Prob.  72a.  Compute  the  head  that  produces  a  velocity  of 
approach  of  50.5  centimeters  per  second. 

Prob.  726.  What  are  the  discharges,  in  liters  per  minute, 
over  a  suppressed  weir  2.35  meters  long  when  the  heads  on  the 
crest  are  12.3,  12.4,  and  12.5  centimeters? 

Prob.  72c.  Compute  the  discharge  over  a  submerged  weir 
when  6  =  2.35,  H  =  0.123,  and  Hr  =  0.027  meters. 

Prob.  72d.  Compute  the  discharge  over  a  dam,  like  Fig. 
686,  when  the  side  slopes  are  i  on  2,  the  length  of  the  crest 
4.25  meters,  and  the  head  on  the  crest  1.07  meters. 


170 


FLOW    THROUGH    TUBES 


CHAP.  VII 


CHAPTER  VII 
FLOW  OF  WATER  THROUGH  TUBES 

ART.  73.     Loss  OF  ENERGY  OR  HEAD 

A  tube  is  a  short  pipe  which  may  be  attached  to  an 
orifice  or  be  used  for  connecting  two  vessels.  The  most 
common  form  is  a  cylinder  of  uniform  cross-section,  but 
conical  forms  are  also  used  and  in  some  cases  a  tube  is 
made  of  cylinders  with  different  diameters.  The  laws 
of  flow  through  tubes  are  important  as  a  starting  point 
for  the  theory  of  flow  through  pipes,  for  the  discharge 
from  nozzles,  and  for  the  discussion  of  many  practical 
hydraulic  problems.  The  theorem  of  Art.  32,  that  pressure- 
head  plus  velocity-head  is  a  constant  for  a  given  section 
of  a  tube,  is  only  true  when  there  are  no  losses  due  to  fric- 
tion and  impact.  As  a  matter  of  fact  such  losses  always 
exist  and  must  be  regarded  in  practical  computations. 

Energy  in  a  tube  filled  with  moving  water  exists  in 
two  forms,  in  potential  energy  of  pressure  and  in  kinetic, 
energy  of  motion.      Thus  in  the  horizontal  tube  of  Fig. 

73a  let  two  piezometers  (Art. 
37)  be  inserted  at  the  sections 
aL  and  a2  where  the  velocities 
are  v^  and  v2  and  it  is  found 
that  the  water  rises  to  the 
heights  h±  and  h2  above  the 
middle  of  the  tube.  Let  W 
be  the  weight  of  water  that 
FlG-  73a  passes  each  section  per  second. 

Then  in  the  first  section  the  pressure  energy  is  Wh^  and 


ART.  73  LOSS    OF   ENERGY    OR   HEAD  171 

the  kinetic  energy  is  W.v12/2g,  so  that  the  total  energy 
of  the  water  passing  that  section  in  one  second  is 


In  the  same  manner  the  total  energy  of  the  water  passing 
the  second  section  in  one  second  is 

Wh2  +  VV.v22/2g 

but  this  is  less  than  the  former  because  some  energy  has 
been  expended  in  friction  and  impact.  Let  Whf  be  the 
amount  of  energy  thus  lost;  then  equating  this  to  the 
difference  of  the  energies  in  the  two  sections,  the  W 
cancels  out  and 

*'-*.-*»+:£H£  (73)> 

The  quantity  h'  is  called  the  lost  head,  and  the  equation 
shows  that  it  equals  the  difference  of  the  pressure-heads 
plus  the  difference  of  the  velocity  -heads. 

In  hydraulics  the  terms  energy  and  head  are  often 
used  as  equivalent,  although  really  energy  is  proportional 
to  head.  In  the  general  case,  the  lost  head  is  not  a  loss 
of  pressure-head  only,  but  a  loss  of  both  pressure-head  and 
velocity-head.  When,  however,  the  two  sections  are  of 
equal  area,  the  velocities  vl  and  v2  are  equal,  since  the  same 
quantity  of  water  passes  each  section  in  one  second; 
then  the  lost  head  hr  is  hl  —  h2  or  the  loss  occurs  in  pressure- 
head  only.  Here  the  loss  is  mainly  due  to  the  roughness 
of  the  interior  surface  of  the  tube  or  pipe.  It  should  be 
noted  that  it  is  only  necessary  to  measure  the  difference 
hl  —  h2  and  this  can  be  done  by  the  methods  of  Art.  37. 

Formula  (73)  t  is  applicable  to  all  horizontal  tubes 
and  pipes,  and  with  a  slight  modification  it  is  also  ap- 
plicable to  inclined  ones,  as  will  be  shown  in  Art.  82. 
It  also  applies  to  a  flow  from  a  standard  orifice,  or  to 
the  flow  from  an  orifice  to  which  a  tube  is  attached.  Thus 


172 


FLOW    THROUGH    TUBES 


CHAP.  VII 


for  the  large  vessel  of  Fig.  736  let  the  sections  be  taken 
through  the  vessel  and  through  the  stream  as  it  leaves 
the  tube.  Then  hi=h,  and  since  there  is  no  pressure 
outside  the  tube  h2  =  o;  also  v1  =  o  and  v 2  =  v ;  then 
h'  =h  —  v2/2g.  For  the  case  in  Fig.  73c,  where  the  stream 
approaches  with  the  velocity  vlt  the  formula  becomes 
h'  =  h1  +  (vl2  —  v2)/2g.  In  both  cases,  if  h'  be  made  zero, 
these  equations  reduce  to  those  established  in  the  chapter 


FIG.  736  FIG.  73c 

on  theoretical  hydraulics,  where  losses  of  energy  were  not 
considered ;  thus  for  the  second  case  the  theoretic  effective 
head  h  is  equal  to  h1  +  vl2/2g. 

In  order  to  use  (73)  t  for  numerical  computations 
three  quantities  must  be  known,  the  difference  h^  —  h^ 
and  the  velocities  v1  and  v2.  As  a  direct  measurement 
of  the  velocities  is  usually  impracticable,  these  are  generally 
computed  from  the  measured  discharge  q  and  the  areas 
aj  and  a2  of  the  cross-sections ;  thus  vv  =  q/a^  and  v2  = 
q/a2.  For  example,  let  the  cross-section  be  circular, 
having  diameters  of  18  and  6  inches,  and  let  the  discharge 
be  4.7  cubic  feet  per  second;  then  from  Table  51  the 
areas  are  1.767  and  0.196  square  feet,  and  the  velocities 
are  2.66  and  23.94  feet  per  second.  If  the  difference 
of  the  pressure-heads  is  8.85  feet  the  lost  head  is 

h'  =  8. 85+0. 01555(2. 662- 23. 942)  =0.05  feet  ' 

The  general  formula  (73)  t  may  be  expressed  in  terms  of 
the  areas  of  the  sections  and  one  of  the  velocities.  Since 
alv1=a2v2,  it  may  be  written 

(73)* 


ART.  74 


EXPANSION  OF  SECTION 


173 


(73). 


which    are    often    more    convenient    forms    for   numerical 
computations. 

Prob.  73a.  When   no   energy   is   lost   between   the   sections 
show  that  the  velocity  v2  is  V2g(h1  —  h2)  +  vl2. 

•  Prob.  736.  In  Fig.  73a  let  the  areas  ax  and  a2  be  i.o  and  0.5 
square  feet,  h1  —  h2  =  o.6<)'j  feet,  and  ^  =  3.5  feet  per  second. 
Show  that  the  lost  head  is  0.126  feet. 


ART.  74.     Loss  DUE  TO  EXPANSION  OF  SECTION 

When  a  tube  or  pipe  is  filled  with  flowing  water  a 
loss  of  head  is  found  to  occur  when  the  section  is  en- 
larged so  that  the  velocity  is  diminished.  This  case  is 
shown  in  Fig.  74a,  where  v^  and  v2  are  the  velocities  in  the 
smaller  and  larger  sections  and 
hi  and  h2  the  corresponding 
pressure-heads.  The  interior 
surface  may  be  very  smooth, 
so  that  friction  has  but  little 
influence,  and  yet  there  will 
usually  be  more  or  less  loss 
due  to  the  fact  that  the  ve- 
locity vl  is  changed  to  the 
smaller  value  v2.  Formula  (73)!  FIG.  74o 

is  here  directly  applicable  and  gives  the  loss  of  head. 
It  is  seen  that  /it  — /22  must  be  negative  for  this  case  and 
that  its  numerical  value  will  be  less  than  that  of  the 
difference  of  the  velocity -heads.  The  general  formula 
(73) !  gives  the  loss  of  head  due  not  only  to  expansion 
of  section,  but  to  all  resistances  between  any  two  sections 
of  a  horizontal  tube  or  pipe. 

When   there   is   a   sudden   enlargement  of  section,    as 
in  Fig.  746,  energy  is  lost  in  impact.     In   the  section  AB 


174 


FLOW   THROUGH    TUBES 


CHAP.  VII 


the  pressure-head  is  h^  and  the  velocity-head  is  v^/2g,  while 
in  the  section  CD  the  pressure-head  has  the  larger  value  h2 

and  the  velocity-head  has  the 
smaller  value  vS/2g.  At  the 
section  MN,  near  the  place  of 
sudden  expansion,  the  pressure- 
head  is  also  hlt  since  the  velocity 
vl  is  maintained  for  a  short 
distance  after  leaving  the  small 
section,  its  direction,  however, 
being  changed  so  as  to  form 
whirls  and  foam.  In  this  region 
the  impact  occurs,  the  velocity 
v1  being  finally  decreased  to  vv  Let  a,  be  the  area  of  the 
sections  MN  and  CD,  and  w  the  weight  of  a  cubic  unit  of 
water.  Then  by  (15)  the  hydrostatic  pressure  normal  to 
the  section  CD  is  iva2h2,  and  that  normal  to  the  section 
MN  is  wajt^  The  difference  of  these  pressures  is  the  force 
which  causes  the  velocity  vl  to  decrease  to  v2,  and  by  Art. 
29  this  force  is  equal  to  W^u^  —  v^/g,  where  W  is  the  weight 
of  water  passing  the  section  CD  in  one  second.  Hence 


D 


FIG.  746 


g 


and,  since  W  equals  wa2v2,  this  equation  becomes 


(74), 


Inserting  this  value  of  h^—h^  in  (73),,  it  reduces  to 


which  is  the  loss  of  head  due  to  sudden  expansion  of  sec- 
tion, or  rather  due  to  the  sudden  diminution  of  velocity 
caused  by  that  expansion. 

If   the    expansion  of  section    be  made  gradually   and 
with  smooth  curves,  the  velocity  vl  will  decrease  without 


ART.  74  EXPANSION    OF   SECTION  175 

whirl  and  foam,  so  that  no  loss  in  impact  occurs.  In  this 
case  the  kinetic  energy  w.vl2/2g  is  changed  into  pressure 
energy,  as  the  velocity  v^  decreases  to  v2.  There  is,  how- 
ever, no  distinct  line  of  demarkation  between  sudden  and 
gradual  expansion,  so  that  in  many  practical  cases  it 
is  necessary  to  make  measurements  of  the  discharge  and 
of  the  head  h^  —  h^  in  order  to  compute  the  lost  head  hr 
from  (73)^  which  is  a  formula  applicable  to  all  cases. 

Sudden  enlargement  of  section  should  always  be 
avoided  in  tubes  and  pipes  owing  to  the  loss  of  head  that 
it  causes,  which  may  often  be  very  great.  For  example, 
let  there  be  no  pressure-head  in  the  section  a^  and  let 
Vi  be  due  to  a  head  h  so  that  vl  =\/2gh  ;  let  the  area  a2 
be  four  times  that  of  at  so  that  v2  is  one-fourth  of  vlt  The 
loss  of  head  due  to  the  sudden  expansion  then  is 


so  that  more  than  one-half  of  the  energy  of  the  water 
in  a1  is  lost  in  impact,  having  been  changed  into  heat. 
In  the  section  a2  the  effective  head  is  &h,  of  which  -fak 
is  velocity  -head  and  TV*  is  pressure-head.  % 

Formula  (74)  ^  may  be  expressed  in  terms  of  the  areas 
of  the  sections  and  one  of  the  velocities,  since  alvl  =  a2v2. 
The  value  of  hf  takes  the  two  forms 

w  } 

/   2g 
\ 

and  these  show  that  no  loss  of  head  occurs  when  ai=a2. 

Prob.  74a.  What  part  of  the  energy  of  the  water  is  lost 
wlien  a2  is  ten  times  a,  ? 

Prob.  746.  In  a  horizontal  tube  like  Fig.  74a  the  diameters 
are  6  inches  and  12  inches,  and  the  heights  of  the  pressure- 
columns  or  piezometers  are  12.16  feet  and  12.96  feet  above 
the  same  bench-mark.  Find  the  loss  of  head  between  the  two 
sections'  when  the  discharge  is  1.57  cubic  feet  per  second,  and 
also  when  it  is  4.71  cubic  feet  per  second. 


176 


FLOW    THROUGH    TUBES 


CHAP.  VII 


ART.  75.     Loss  DUE  TO  CONTRACTION  OF  SECTION 

When  a  sudden  contraction  of  section  in  the  direction 
of  the  flow  occurs,  as  in  Fig.  75,  the  water  suffers  a  con- 
traction similar  to  that  in  the  standard  orifice,  and  hence 
in  its  expansion  to  fill  the  second 
section  a  loss  of  head  results.  Let 
v^  be  the  velocity  in  the  larger 
section  and  v  that  in  the  smaller, 
while  vf  is  the  velocity  in  the  con- 
tracted section  of  the  flowing  stream  ; 
and  let  alt  a,  and  a'  be  the  corre- 
sponding areas  of  the  cross-sections. 
From  the  formula  (74)  2  the  loss 
of  head  due  to  the  expansion  of 


FIG.  75 


section  from  a!  to  a  is 

\2^2  /j  \2V3 

—  =    -7-  i  I   —  (75)j 

2g       V          I     2g 

in  which  cf  is  the  coefficient  of  contraction  of  the  stream 
or  the  ratio  of  a'  to  a  (Art.  44). 

The  value  of  cr  depends  upon  the  ratio  between  the 
areas  a  and  alt  When  a  is  small  compared  with  alt  the 
value  of  c'  may  be  taken  at  0.62  as  for  orifices  (Art.  44). 
When  a  is  equal  to  aA  there  is  no  contraction  or  expansion 
of  the  stream  and  cf  is  unity.  Let  d  and  d^  be  the  diam- 
eters corresponding  to  the  areas  a  and  alf  and  let  r  be 
the  ratio  of  d  to  dlt  Then  experiments  seem  to  indicate 
that  an  expression  of  the  form 

n 

c'  =  m  +  - 

i.i—  r 

gives  the  law  of  variation  of  c'  with  r.  Placing  cf  =0.62 
and  r  =  o  gives  one  equation  between  m  and  n\  placing 
c'  =  1.00  and  r  =  i  gives  another  equation;  and  the  solution 
of  these  furnishes  the  values  of  m  and  n.  Thus  is  found 

0.0418 


£'=0.582 


i.i— r 


(75), 


ART.  75  CONTRACTION    OF   SECTION  177 

from  which  approximate  values  of  cf  can  be  computed; 

for       r  =  o.o      0.4      0.6      0.7      0.8      0.9      0-95    i.o 
£'=0.62    0.64    0.67    0.69    0.72    0.79    0.86     i. oo 

and  from  these  intermediate  values  may  often  be  taken 
without  the  necessity  of  using  the  formula. 

For  a  case  of  gradual  contraction  of  section,  such  as 
shown  in  Fig.  73a,  the  loss  of  head  is  less  than  that  given 
by  formula  (74)  v  and  it  can  only  be  determined  from 
three  measured  quantities  by  the  help  of  the  general 
formulas  of  Art.  73.  If  the  change  of  section  be  made 
so  that  the  stream  has  no  subsequent  enlargement,  loss 
of  head  is  avoided,  for,  as  the  above  discussions  show, 
it  is  the  loss  in  velocity  due  to  sudden  expansion  which 
causes  the  loss  of  head. 

The  loss  due  to  sudden  contraction  of  a  tube  or  pipe 
is  often  much  smaller  than  that  due  to  sudden  expansion. 
For  instance,  if  the  diameter  of  the  large  section  be  three 
times  that  of  the  smaller,  and  the  velocity  in  the  large 
section  be  2  feet  per  second,  the  loss  of  head  when  the 
flow  passes  from  the  small  to  the  large  section  is  by  Art.  74 

hf  =0.01555(18  —  2)2=4.o  feet 

But  if  the  flow  occurs  in  the  opposite  direction  the  ratio  r 
is  J,  the  coefficient  c'  is  about  0.64,  and  the  loss  of  head  is 

/    j          \2 
&'±=o. 01555!  — 7 i)    1 82  =  1.6  feet 

When,  however,  the  ratio  r  is  higher  than  0.77,  the  loss 
due  to  sudden  contraction  is  greater  than  that  due  to 
sudden  expansion.  Thus,  if  the  diameter  of  the  small 
section  be  nine-tenths  that  of  the  large  one  and  the  ve- 
locity in  the  large  section  be  2  feet  per  second,  the  loss  of 
head  when  the  flow  passes  from  the  small  to  the  large 

section  is 

/    T         \2 
hr  =  o.oi^"d — 5— —  i)    22  =  0.0034  feet 

).OI  / 


178  FLOW  THROUGH  TUBES  CHAP,  vn 

But  if  the  flow  occurs  in  the  opposite  direction  the  ratio 
r  is  0.9,  the  coefficient  c'  is  0.79,  and  the  loss  of  head  is 

/i       V 

h'  =  o. 01555! 1)  2.472  =  o.oo66  feet 

As  formula  (75)2  is  an  empirical  one  the  results  derived 
from  it  are  to  be  regarded  as  approximate. 

Prob.  75a.  Show  that  the  loss  due  to  sudden  contraction 
is  the  same  as  that  due  to  sudden  expansion  when  the  ratio 
r  is  equal  to  0.77. 

Prob.  756.  Compute  the  loss  of  head  when  a  pipe  which  dis- 
charges 1.57  cubic  feet  per  second  suddenly  diminishes  in  section 
from  12  to  6  inches  in  diameter. 


ART.  76.     THE  STANDARD  SHORT  TUBE 

An  adjutage  is  a  tube  inserted  into  an  orifice,  and  the 
short-tube  adjutage,  consisting  of  a  cylinder  whose  length 
is  about  three  times  its  diameter,  is  the  most  common 
form.  For  convenience  it  will  be  called  the  standard 
short  tube,  because  its  theory  and  coefficients  form  a 
starting  point  with  which  all  other  adjutages  may  be 
compared.  This  short  tube  is  of  little  value  for  the 
measurement  of  water,  since  the  coefficients  for  standard 
orifices  are  much  more  definitely  known.  The  discussion 
here  given  is  for  the  case  where  the  inner  edge  is  a  sharp, 
definite  corner  like  that  of  the  standard  orifice  (Art.  43). 
If  the  tube  be  only  two  diameters  in  length  the  stream 

passes  through  without 
touching  it,  as  in  the 
first  diagram  of  Fig. 
76>  and  the  discharge 
is  the  same  as  from  the 
orifice.  If  it  be  length- 
ened sufficiently  the  stream  expands  and  fills  the  tube,  as  in 
the  second  diagram,  and  the  discharge  is  much  increased.  By 


ART.  76  THE    STANDARD    SHORT   TUBE  179 

observations  on  glass  tubes  it  is  seen  that  the  stream 
usually  contracts  after  leaving  the  inner  end  of  the  tube 
and  then  expands.  This  contraction  may  be  apparently 
destroyed  by  agitating  the  water  or  by  striking  the  tube, 
and  the  entire  tube  is  then  filled,  yet  if  a  hole  be  bored 
in  the  tube  near  its  inner  end  water  does  not  flow  out, 
but  air  enters,  showing  that  a  negative  pressure  exists. 

An  estimate  of  the  velocity  and  discharge  from  this 
short-tube  adjutage  may  be  made  as  follows:  Let  h  be 
the  head  on  the  inner  end  of  the  tube  and  v  the  velocity 
of  the  outflowing  water.  The  head  h  equals  the  velocity- 
head  v2/2g  plus  all  the  losses  of  head.  At  the  inner  edge 
a  loss  of  o.o4V2/2g  occurs  in  entering  the  tube,  as  in  the 
standard  orifice  (Art.  56),  and  then  there  is  a  loss  of 
(v'  —  v)2/2g  when  the  contracted  stream  suddenly  expands 
so  that  its  velocity  v'  is  reduced  to  v  (Art.  74).  If  a' 
and  a  be  the  areas  of  these  two  sections,  their  ratio  a' '/a 
is  the  coefficient  of  contraction  c' '.  Then 

V2       /I  \2V2        V2 

h  =  0.04 —  -f  (  -7  —  i  I 1 

42g^V  /    2g       2g 

Now,  taking  for  cr  its  mean  value  0.62,  this  equation 
reduces  to  v  =  o.%4\/2gh,  or  the  coefficient  of  velocity 
of  the  issuing  jet  is  0.84.  Since  the  cross-section  of  the 
stream  at  the  outer  end  of  the  tube  is  the  same  as  that 
of  the  tube,  the  coefficient  of  contraction  for  that  end  is 
unity,  and  hence  (Art.  46)  the  coefficient  of  discharge 
is  also  0.84. 

Experiments  indicate,  however,  that  this  coefficient 
is  too  large,  and  this  is  to  be  expected,  since  the  above 
investigation  does  not  include  the  loss  due  to  friction 
along  the  sides  of  the  tube  after  the  stream  has  expanded. 
From  the  experiments  of  Venturi  and  Bossut  it  appears 
that  a  mean  value  is 

Coefficient  of  discharge  £  =  0.82 


180 


FLOW    THEOUGH    TUBES 


CHAP.  VII 


This  coefficient,  however,  ranges  from  0.83  for  low  heads 
to  0.79  for  high  heads.  It  is  greater  for  large  tubes  than 
for  small  ones,  its  law  of  variation  being  probably  the  same 
as  for  orifices  (Art.  47),  but  sufficient  experiments  have 
not  been  made  to  state  definite  values  in  the  form  of  a 
table. 

A  standard  orifice  gives  on  the  average  about  61  per- 
cent of  the  theoretic  discharge,  but  by  the  addition  of  a 
tube  this  may  be  increased  to  82  percent.  The  velocity- 
head  of  the  jet  from  the  tube  is,  however,  much  less  than 
that  from  the  orifice.  For,  let  v  be  the  velocity  and  h 
the  head,  then  (Art.  45)  for  the  standard  orifice 


or 


and  similarly  for  the  standard  tube 

or 

Accordingly  the   velocity-head   of    the    stream  from   the 

standard  orifice  is  96  per- 
cent of  the  theoretic  ve- 
locity-head, and  that  of 
the  stream  from  the  stand- 
ard tube  is  only  67  per- 
Cent.  Or  if  jets  be  directed 
vertically  upward  from  a 
standard  orifice  and  tube, 
as  in  Fig.  76c,  that  from 
the  former  rises  to  the 
FlG-76c  height  0.96/2,  while  that 

from  the  latter  rises  to  the  height  0.67/2,  where  h  is  the 
head  from  the  level  of  water  AB  in  the  reservoir  to  the 
point  of-  exit. 

The  energy  lost  in  the  stream  from  the  standard  orifice 
is  hence  4  percent  of  the  theoretic  energy,  but  in  that 
from  the  standard  tube  33  percent  is  lost.  In  reality 
energy  is  never  lost,  but  is  merely  transformed  into  other 


ART.  76          THE  STANDARD  SHORT  TUBE  181 

forms  of  energy.  In  the  tube  the  one-third  of  the  total 
energy  which  has  been  called  lost  is  only  lost  because  it 
cannot  be  utilized  as  work;  it  is,  in  fact,  transformed 
into  heat,  which  raises  the  temperature  of  the  water. 
The  above  explanation  shows  that  most  of  this  loss  is 
due  to  the  impact  resulting  from  the  sudden  expansion 
of  the  stream. 

The  loss  of  head  in  the  flow  from  the  short  tube  is 
large,  but  not  so  large  as  might  be  expected  from  theoretical 
considerations  based  on  the  known  coefficients  for  orifices. 
If  the  tube  has  a  length  of  only  two  diameters  the  water 
does  not  touch  its  inner  surface,  and  the  flow  occurs  as 
from  a  standard  orifice.  The  velocity  in  the  plane  of 
the  inner  end  is  then  61  percent  of  the  theoretic  velocity, 
since  the  mean  coefficient  of  discharge  is  0.6 1.  -Now  when 
the  tube  is  sufficiently  increased  in  length  its  outer  end 
will  be  filled,  and  if  the  contraction  still  exists,  it  might 
be  inferred  that  the  coefficient  for  that  end  would  be 
also  0.6 1 ;  this  would  give  a  velocity-head  of  (0.6 i)2/2  or 
0.37/2,  so  that  the  loss  of  head  would  be  0.63/2.  Actually, 
however,  the  coefficient  is  found  to  be  0.82  and  the  loss 
of  head  only  0.33/2.  It  hence  appears  that  further  ex- 
planation is  needed  to  account  for  the  increased  discharge 
and  energy. 

In  the  first  place,  a  loss  of  about  0.04/2  occurs  at  the 
inner  end  of  the  tube  in  the  same  manner  as  in  the  stand- 
ard orifice,  and  only  the  head  0.96/2  is  then  available  for 
the  subsequent  phenomena.  If  the  coefficient  cf  for  the 
contracted  section  have  the  value  0.62,  the  velocity  in 
that  section  is 

,     0.82    / — r  / — r 

V  = — -7~ V  2gk  =  1. 32V  2gk 

and  the  velocity-head  for  that  section  is 


182 


FLOW    THROUGH    TUBES 


CHAP.  VII 


and  consequently  the  pressure-head  in  that  section  is 
0.96/2— 1.75^  =  — o.yg/i 

There  exists  therefore  a  negative  pressure  or  partial 
vacuum  near  the  inner  end  of  the  tube  which  is  sufficient 
to  lift  a  column  of  water  to  a  height 
of  about  three-fourths  the  head. 
This  conclusion  has  been  confirmed 
by  experiment  for  low  heads,  and 
was  in  fact  first  discovered  ex- 
perimentally by  Venturi.  For  high 
heads  it  is  not  valid,  since  in  no 
event  can  atmospheric  pressure  raise 
a  column  of  water  higher  than 
about  34  feet  (Art.  5) ;  probably 
under  high  heads  the  coefficient 
of  contraction  of  the  stream  in 
the  tube  becomes  much  greater  than  0.62. 

The  cause  of  the  increased  discharge  of  the  tube  over 
the  orifice  is  hence  a  partial  vacuum,  which  causes  a  portion 
of  the  atmospheric  head  of  34  feet  to  be  added  to  the  head 
h,  so  that  the  flow  at  the  contracted  section  occurs  as  if 
under  the  head  h  +  h^.  The '  occurrence  of  this  partial  vac- 
uum is  attributed  to  the  friction  of  the  water  on  the  air. 
When  the  flow  begins,  the  stream  is  surrounded  by  air 
of  the  normal  atmospheric  pressure  which  is  imprisoned 
as  the  stream  fills  the  tube.  The  friction  of  the  moving 
water  carries  some  of  this  air  out  with  it,  thus  rarefying 
the  remaining  air.  This  rarefaction,  or  negative  press- 
ure, is  followed  by  an  increased  velocity  of  flow,  and  the 
process  continues  until  the  air  around  the  contracted 
section  is  so  rarefied  that  no  more  is  removed,  and  the 
flow  then  remains  permanent,  giving  the  results  ascertained 
by  experiment.  The  partial  vacuum  causes  neither  a 
gain  nor  loss  of  head,  for  although  it  increases  the  velocity- 
head  at  the  contracted  section  to  1.75/1,  there  must  be 


ART.  77  CONICAL    CONVERGING   TUBES  183 

expended  0.79/2  in  order  to  overcome  the  atmospheric 
pressure  at  the  outer  end  of  the  tube.  The  experiments 
of  Buff  have  proved  that  in  an  almost  complete  vacuum 
the  discharge  of  the  tube  is  but  little  greater  than  that 
of  the  orifice.* 

Prob.  76.  If  the  coefficient  of  contraction  for  the  contracted 
section  is  0.70,  show  that  the  probable  coefficient  of  discharge 
is  about  0.90.  Also  show  that,"  for  these  data,  the  negative 
pressure-head  is  about  o.yo/t. 

ART.  77.     CONICAL  CONVERGING  TUBES 

Conical  converging  tubes  are  used  when  it  is  desired 
to  obtain  a  high  efficiency  in  the  energy  of  the  stream 
of  water.  At  A  is  shown 
a  simple  converging  tube, 
consisting  of  a  frustum  of  f^ 
a  cone,  and  at  B  is  a 
similar  frustum  provided  ~~.'\A 
with  a  cylindrical  tip.  The 
proportions  of  these  con- 
verging tubes,  or  mouthpieces,  vary  somewhat  in  practice, 
but  the  cylindrical  tip  when  employed  is  of  a  length  equal 
to  about  2\  times  its  inner  diameter,  while  the  conical 
part  is  eight  or  ten  times  the  length  of  that  diameter, 
the  angle  at  the  vertex  of  the  cone  being  between  10  and 
20  degrees. 

The  stream  from  a  conical  converging  tube  like  A 
suffers  a  contraction  at  some  distance  beyond  the  end. 
The  coefficient  of  discharge  is  higher  than  that  of  the 
standard  tube,  being  generally  between  0.85  and  0.95, 
while  the  coefficient  of  velocity  is  higher  still.  Experi- 
ments made  by  d'Aubuisson  and  Castel  on  conical  con- 
verging tubes  0.04  meters  long  and  0.0155  meters  in 
diameter  at  the  small  end,  under  a  head  of  3  meters,  ftir- 

*  Annalen  der  Physik  und  Chemik,  1839,  vol.  46,  p.  242. 


184  FLOW  THROUGH  TUBES  CHAP,  vii 

nish  the  coefficients  of  discharge  and  velocity  given  in 
Table  31.  The  former  of  these  was  determined  by  measur- 
ing the  actual  discharge  (Art.  46),  and  the  latter  by  the 
range  of  the  jet  (Art.  45).  The  coefficient  of  contraction 
as  computed  from  these  is  given  in  the  last  column,  and 
this  applies  to  the  jet  at  the  smallest  section,  some  dis- 
tance beyond  the  end  of  the  tube.  While  these  values 
show  that  the  greatest  discharge  occurred  for  an  angle 
of  about  13^  degrees,  they  also  indicate  that  the  coefficient 
of  velocity  increases  with  the  convergence  of  the  cone, 
becoming  about  equal  to  that  of  a  standard  orifice  for 
the  last  value.  Hence  the  table  seems  to  teach  that  a 
conical  frustum  does  not  usually  give  as  high  a  velocity  as 
a  standard  orifice. 

Under  very  high  heads,  over  300  feet,  Hamilton  Smith 
found  the  actual  discharge  to  agree  closely  with  the  theo- 
retical, or  the  coefficient  of  discharge  was  nearly  i.o,  and 
in  some  case  slightly  greater.*  His  tubes  were  about 
0.9  feet  long,  o.i  feet  in  diameter  at  the  small  end  and 
0.35  feet  at  the  large  end,  the  angle  of  convergence  being 
17  degrees.  As  these  figures  indicate  a  contraction  of 
the  jet  beyond  the  end,  it  cannot  be  supposed  that  the 
coefficient  of  discharge  in  any  case  was  really  as  high  as 
his  experiments  indicate.  Under  these  high  heads  the 
cylindrical  tip  applied  to  the  end  of  a  tube  produced  no 
effect  on  the  discharge,  the  jet  passing  through  without 
touching  its  surface. 

Prob.  77.  If  the  coefficient  of  discharge  is  0.98  and  the 
coefficient  of  velocity  0.995,  compute  the  coefficient  of  con- 
traction. 

ART.  78.     INWARD  PROJECTING  TUBES 

Inward  projecting  tubes,  as  a  rule,  give  a  less  dis- 
charge than  those  whose  ends  are  flush  with  the  sides 
of  the  reservoir,  due  to  the  greater  convergence  of 

*  Smith's  Hydraulics,  p.  286. 


ART.  78  INWARD    PROJECTING    TUBES  185 

the  lines  of  direction  of  the  filaments  of  water.  At  A 
and  B  are  shown  inward  projecting  tubes  so  short  that 
the  water  merely  touches  their  inner  edges,  and  hence 
they  may  more  properly  be  called  orifices.  Experiment 
shows  that  the  case  at  A,  where  the  sides  of  the  tube 


B=S«W»^    =^^=_=^   =l€&®m= 


FIG.  78 

are  normal  to  the  side  of  the  reservoir,  gives  the  minimum 
coefficient  of  discharge  c  =  0.5,  while  for  B  the  value  lies 
between  0.5  and  that  for  the  standard  orifice  at  C.  The 
inward  projecting  cylindrical  tube  at  D  has  been  found  to 
give  a  discharge  of  about  72  percent  of  the  theoretic  dis- 
charge, while  the  standard  tube  (Art.  76)  gives  82  percent. 
For  the  tubes  E  and  F  the  coefficients  depend  upon  the 
amount  of  inward  projection,  and  they  are  much  larger 
than  0.72  for  both  cases,  when  computed  for  the  area  of  the 
smaller  end. 

It  is  usually  more  convenient  to  allow  a  water-main  to 
project  inward  into  the  reservoir  than  to  arrange  it  with 
its  mouth  flush  to  a  vertical  side.  The  case  D,  in  Fig. 
78,  is  therefore  of  practical  importance  in  considering  the 
entrance  of  water  into  the  main.  As  the  end  of  such  a 
main  has  a  flange,  forming  a  partial  bell-sha'ped  mouth, 
the  value  of  c  is  probably  higher  than  0.72.  The  usual 
value  taken  is  0.82,  or  the  same  as  for  the  standard  tube. 
Practically,  as  will  be  seen  later,  it  makes  little  difference 
which  of  these  is  used,  as  the  velocity  in  a  water-main 
is  slow  and  the  resistance  at  the  mouth  is  very  small  com- 
pared with  the  frictional  resistances  along  its  length. 

Prob.  78.  Find  the  coefficient  of  discharge  for  a  tube  whose 
diameter  is  one  inch  when  the  flow  under  a  head  of  9  feet  is 
22.1  cubic  feet  in  3  minutes  and  30  seconds. 


186 


FLOW    THROUGH    TUBES 


CHAP.  VII 


FIG.  79 


ART.  79.     DIVERGING  AND  COMPOUND  TUBES 

In  Fig.  79  is  shown  a  diverging  conical  tube,  BC,  and 
two  compound  tubes.  The  compound  tube  ABC  consists 
of  two  cones,  the  converging  one,  A B,  being  much  shorter 

than  the  diverging  one,  BC, 
so  that  the  shape  roughly 
approximates  to  the  form  of 
the  contracted  jet  which  is- 
sues from  an  orifice  in  a 
thin  plate.  In  the  tube  AE 
the  curved  converging  part 
AB  closely  imitates  the  con- 
tracted jet,  and  BB  is  a 
short  cylinder  in  which  all 
the  filaments  of  the  stream 
are  supposed  to  move  in  lines 
parallel  to  the  axis  of  the  tube,  the  remaining  part  being 
a  frustum  of  a  cone.  The  converging  part  of  a  com- 
pound tube  is  often  called  a  mouthpiece  and  the  diverging 
part  an  adjutage. 

Many  experiments  with  these  tubes  have  shown  the 
interesting  fact  that  the  discharge  and  the  velocity  through 
the  smallest  section,  B,  are  greater  than  those  due  to 
the  head;  or,  in  other  words,  that  the  coefficients  of  dis- 
charge and  velocity  for  this  section  are  greater  than  unity. 
One  of  the  first  to  notice  this  was  Bernoulli  in  1738,  who 
found  c  =  i.o&  for  a  diverging  tube.  Venturi  in  1791 
experimented  on  such  tubes,  and  showed  that  the  angle 
of  the  diverging  part,  as  also  its  length,  greatly  influenced 
the  discharge.  He  concluded  that  c  would  have  a  maxi- 
mum value  of  1.46  when  the  length  of  the  diverging  part 
was  9  times  its  least  diameter,  the  angle  at  the  vertex  of  the 
cone  being  5°  06'.  Eytelwein  found  c  =  i.i8  fora  diverg- 
ing tube  like  BC  in  Fig.  79,  but  when  it  was  used  as  an 


ART.  79  DIVERGING    AND    COMPOUND    TUBES  187 

adjutage  to  a  mouthpiece  A B,  thus  forming  a  compound 
tube  ABC,  he  found  ^  =  1.55. 

The  experiments  of  Francis  in  1854  on  a  compound 
tube  like  ABCDE  are  very  interesting.*  The  curve  of 
the  converging  part  AB  was  a  cycloid,  BB  was  a  cylinder, 
and  the  diameters  at  A,  B,  C,  D,  and  E  were  1.4,  0.102, 
0.145,  °-234>  and  °-321  fe^  The  piece  BB  was  o.-i  feet 
long,  and  the  others  each  i  foot;  these  were  made  to 
screw  together,  so  that  experiments  could  be  made  on 
different  lengths.  A  sixth  piece,  EF,  not  shown  in  the 
figure,  was  also  used,  which  was  a  prolongation  of  the 
diverging  cone,  its  largest  diameter  being  0.4085  feet. 
The  tubes  were  of  cast  iron,  and  quite  smooth.  The 
flow  was  measured  with  the  tubes  submerged,  and  the 
effective  head  varied  from  about  o.oi  to  1.5  feet.  Ex- 
cluding heads  less  than  o.i  feet,  the  following  shows  the 
range  in  value  of  the  coefficients  of  discharge: 

c  for  Section  BB.  c  for  Outer  End. 

for  tube  AB,  0.80  to  0.94  0.80  to  0.94 

for  tube  AC,  1.43  to  1.59  0.70  to  0.78 

for  tube  AD,  1.98  to  2.16  0.37  to  0.41 

for  tube  AE,  2.08  to  2.43  0.21  to  0.24 

for  tube  A  E,  2.05  to  2.42  0.13  to  0.15 

The  maximum  discharge  was  thus  found  to  occur  with 
the  tube  AE,  and  to  be  2.43  times  the  theoretic  discharge 
that  would  be  expected  for  the  small  section  BB.  In 
general  the  coefficients  increased  with  the  heads,  the  value 
2.08  being  for  a  head  of  0.13  feet  and  2.43  for  a  head  of 
1.36  feet;  for  1.39  feet,  however,  c  was  found  to  be  2.26. 

These  coefficients  of  discharge  are  the  same  as  the 
coefficients  of  velocity,  since  the  tube  was  entirely  'filled. 
Thus,  when  the  coefficient  for  the  section  BB  was  2.43 
the  velocity  was  v  =  2.^\/2gh,  and  the  velocity -head  was 

^2/2g  =  (2.43)2/*  =  5.9o/* 

*  Lowell  Hydraulic  Experiments,  4th  Edition,  pp.  209-232. 


188  FLOW  THROUGH  TUBES  CHAP,  vii 

Therefore  the  flow  through  the  section  BB  was  that  due 
to  a  head  5.9  times  greater  than,  the  actual  head  of  1.36 
feet;  or,  in  other  words,  the  energy  of  the  water  flowing 
in  BB  was  5.9  times  the  theoretic  energy.  Here,  ap- 
parently, is  a  striking  contradiction  of  the  fundamental 
law  of  the  conservation  of  energy.  The  explanation  of 
this  apparent  contradiction  is  the  same  as  that  given 
in  Art.  76  for  the  short  tube  adjutage.  The  increased 
velocity  and  discharge  is  due  to  the  occurrence  of  a  par- 
tial vacuum  near  the  inner  end  of  the  adjutage  BC.  The 
pressure  of  the  atmosphere  on  the  water  in  the  reservoir 
thus  increases  the  hydrostatic  pressure  due  to  the  head, 
and  the  increased  flow  results.  The  energy  at  the  smallest 
section  is  accordingly  higher  than  the  theoretic  energy. 
but  the  excess  of  this  above  that  due  to  the  head  must 
be  expended  in  overcoming  the  atmospheric  pressure 
on  the  outer  end  of  the  tube,  so  that  in  no  case  does  the 
available  exceed  the  theoretic  energy.  No  contradiction 
of  the  law  of  conservation  therefore  exists. 

To  render  this  explanation  more  definite,  let  the  ex- 
treme case  be  considered  where  a  complete  vacuum  exists 
near  the  inner  end  of  the  adjutage,  if  that  were  possible, 
as  it  perhaps  might  be  with  a  tube  of  a  certain  form. 
Let  h  be  the  head  of  water  in  feet  on  the  center  of  the 
smallest  section.  The  mean  atmospheric  pressure  on  the 
water  in  the  reservoir  is  equivalent  to  a  head  of  34  feet 
(Art.  5).  Hence  the  total  head  which  causes  the  discharge 
into  the  vacuum  is  /£  +  34  and  the  velocity  of  flow  is  nearly 
Neglecting  the  resistances,  which  are  very 


slight  if  the  entrance  be  curved,  the  coefficients  of  velocity 
and  discharge  can  now  be  found;   thus: 


for  h  =  100, 
iorh=    10, 


35 


ART.  80  NOZZLES   AND    JETS  189 

The  coefficient  hence  increases  as  the  head  decreases. 
That  this  is  not  the  case  in  the  above  experiments  is 
undoubtedly  due  to  the  fact  that  the  vacuum  was  only 
partial,  and  that  the  degree  of  rarefaction  varied  with 
the  velocity.  The  cause  of  the  vacuum,  in  fact,  is  to  be 
attributed  to  the  velocity  of  the  stream,  which  by  fric- 
tion removes  a  part  of  the  air  from  the  inner  end  of  the 
adjutage. 

It  follows  from  this  explanation  that  the  phenomena 
of  increased  discharge  from  a  compound  tube  could  not 
be  produced  in  the  absence  of  air.  The  experiment  has 
been  tried  on  a  small  scale  under  the  receiver  of  an  air- 
pump,  and  it  was  found  that  the  actual  flow  through 
the  narrow  section  diminished  the  more  complete  the  rare- 
faction. It  also  follows  that  it  is  useless  to  state  any 
value  as  representing,  even  approximately,  the  coefficient 
of  discharge  for  such  tubes. 

Prob.  79.  Compute  the  pressure  per  square  inch  in  the  sec- 
tion BB  of  Francis'  tube  when  /*=  1.36  feet  and  £  =  2.43.  What 
is  the  height  of  the  column  of  water  that  can  be  lifted  by  a  small 
pipe  inserted  at 


ART.  80.     NOZZLES  AND  JETS 

For  fire  service  two  forms  of  nozzles  are  in  use.  The 
smooth  nozzle  is  essentially  a  conical  tube  like  A  in  Fig. 
77,  the  larger  end  being  attached  to  a  hose,  but  it  is  often 
provided  with  a  cylindrical  tip  and  sometimes  the  larger 
end  is  curved  as  shown  in  Fig.  80a.  The  ring  nozzle  is 


FIG.  80a  FIG.  806 

a  similar  tube,  but  its  end  is  contracted  so  that  the  water 
issues  through  an  orifice  smaller  than  the  end  of  the  tube. 
The  experiments  of  Freeman  show  that  the  mean  coefficient 


190  FLOW  THROUGH  TUBES  CHAP,  vir 

of  discharge  is  about  0.97  for  the  smooth  nozzle  and  about 
0.74  for  the  ring  nozzle.*  The  smooth  nozzle  is  used 
much  more  than  the  ring  nozzle. 

Let  d  be  the  diameter  of  the  pipe  or  hose  and  D  the 
diameter  of  the  outlet  at  the  end  of  the  nozzle,  and  let 
v  and  V  be  the  corresponding  velocities.  Let  h1  be  the 
pressure-head  at  the  entrance  to  the  nozzle;  then  the 
effective  head  at  the  entrance  to  the  nozzle  is 


and  the  velocity  at  the  end  of  the  nozzle  is  V  =  c1\/2gHr 
where  c1  is  the  coefficient  of  velocity.  The  reasoning  of 
Art.  51  applies  here,  if  the  ratio  D2/d2  be  used  in  place 
of  a/  A,  and  ht  in  place  of  h,  and  hence 


(80)' 

is  the  velocity  of  flow  from  the  nozzle,  c  being  the  co- 
efficient of  discharge.  The  discharge  per  second  is,  from 
formula  (51)2 

(80)- 


The  effective  head  at  the  nozzle  entrance  is 

rr     .j_y!_ 
~ 


and  the  velocity-head  of  the  issuing  jet  is 


which  gives  the  height  to  which  the  jet  would  rise  if  there 
were  no  atmospheric  resistances.     In  these  formulas  D/d 

*  Transactions  American  Society  of  Civil  Engineers,  1889,  vol.  21,  pp. 
303-482. 


ART.  80  NOZZLES    AND  JETS  191 

is  an  abstract  number  and  to  find  its  value  D  and  d  may 
be  taken  in  any  unit  of  measure. 

When  hi  and  D  are  in  feet,  g  is  to  be  taken  as  32.16 
feet  per  second  per  second.  Then  (80)  t  gives  V  in  feet 
per  second  and  (80)  2  gives  q  in  cubic  feet  per  second. 
When  the  gage  at  the  nozzle  entrance  gives  the  pressure 
pi  in  pounds  per  square  inch,  hi  in  feet  is  found  from 
2.$o4pi.  It  is  a  common  practice  in  figuring  on  fire- 
streams  to  compute  the  discharge  in  gallons  per  minute. 
For  this  case,  if  D  be  taken  in  inches, 


gives  the  discharge  in  gallons  per  minute. 

For  smooth  nozzles  the  value  of  the  coefficient  of 
velocity  c^  is  the  same  as  that  of  the  coefficient  of  dis- 
charge c,  since  the  jet  issues  without  contraction.  The 
experiments  of  Freeman  furnish  the  following  mean 
values  of  the  coefficient  of  discharge  for  smooth  cone 
nozzles  of  different  diameters  under  pressure-heads  rang- 

ing from  45  to  180  feet: 

• 

Diameter  in  inches  =      f          |  i  i|  i\          ij 

Coefficient  c  =0.983     0.982     0.972     0.976     0.971     0.959 

These  values  were  determined  by  measuring  the  pressure 
pi  and  the  discharge  q,  from  which  c  can  be  computed 
by  the  last  formula.  For  example,  a  nozzle  having  a 
diameter  of  i.ooi  inches  at  the  end  and  2.50  inches  at 
the  base  discharged  208.5  gallons  per  minute  under  a 
pressure  of  50  pounds  per  square  inch  at  the  entrance. 
Here  Z>  =  i.ooi,  ^  =  2.5,^=50,  and  9  =  208.  5,  and  in- 
serting these  in  the  formula  and  solving  for  c,  there  is 
found  c  =  0.985. 

In  ring  nozzles  the  ring  which  contracts  the  entrance 
is  usually  only  TV  or  \  inch  in  width.  The  effect  of  this 
is  to  diminish  the  discharge,  but  the  stream  is  sometimes 


192  FLOW   THROUGH    TUBES  CHAP.  VII 

thrown  to  a  slightly  greater  height.  On  the  whole,  ring 
nozzles  seem  to  have  no  advantage  over  smooth  ones 
for  fire  purposes.  As  the  stream  contracts  after  leaving 
the  nozzle,  the  coefficient  of  velocity  cl  is  greater  than 
the  coefficient  of  discharge  c.  The  value  of  c  being  about 
0.74,  that  of  cl  is  probably  a  little  larger  than  0.97.  In 
using  (80)  t  for  ring  nozzles  these  values  of  cv  and  c  should 
be  inserted,  but  in  using  (80)2  only  the  value  of  c  is  needed. 

According  to  Freeman's  experiments,  the  discharge  of 
a  f -inch  ring  nozzle  is  the  same  as  that  of  a  f-inch  smooth 
nozzle,  while  the  discharge  of  a  ij-inch  ring  nozzle  is 
about  20  percent  greater  than  that  of  a  i-inch  smooth 
nozzle.  The  heights  of  vertical  jets  from  a  ij-inch  ring 
nozzle  are  about  the  same  as  those  from  a  i-inch  smooth 
nozzle,  while  the  jets  from  a  if -inch  ring  nozzle  are  slightly 
less  in  height  than  those  from  a  ij-inch  smooth  nozzle. 

The  vertical  height  of  a  jet  from  a  nozzle  is  very  much 
less,  on  account  of  the  resistance  of  the  air,  than  the  value 
deduced  above  for  V2/2g.  For  instance,  let  a  smooth 
nozzle  one  inch  in  diameter  attached  to  a  2. 5 -inch  hose 
have  £  =  0.97  and  the  pressure-head  ^  =  230  feet;  then 
the  computation  gives  the  velocity -head  V2/2g  as  221 
feet,  whereas  the  average  of  the  highest  drops  in  still  air 
will  be  about  152  feet  high  and  the  main  body  of  water 
will  be  several  feet  lower.  Table  32,  compiled  from  the 
results  of  Freeman's  experiments,  shows  for  three  different 
smooth  nozzles  the  height  of  vertical  jets,  column  A 
giving  the  heights  reached  by  the  average  of  the  highest 
drops  in  still  air,  and  column  B  the  maximum  limits  of 
height  as  a  good  effective  fire-stream  with  moderate 
wind.  The  discharges  given  depend  only  on  the  pressure, 
and  are  the  same  for  horizontal  as  for  vertical  jets. 

The  maximum  horizontal  distance  to  which  a  jet  can 
be  thrown  is  also  a  measure  of  the  efficiency  of  a  nozzle. 
The  following,  taken  from  Freeman's  tables,  gives  the 


ART.  80  NOZZLES    AND    JETS  193 

horizontal  distances  at  the  level  of  the  nozzle  reached  by 
the  average  of  the  extreme  drops  in  still  air: 

Pressure  at  nozzle  entrance,  20  40  60  80  100  pounds. 

From  f-inch  smooth  nozzle,  72  112  136  153  167  feet. 

From  i -inch  smooth  nozzle,  77  133  167  189  205  feet. 

From  i  ^-inch  smooth  nozzle,  83  148  186  213  236  feet. 

From  i  |-inch  ring  nozzle,  76  131  164  186  202  feet. 

From  i \ -inch  ring  nozzle,  78  138  172  196  215  feet. 

From  i  f-inch  ring  nozzle,  79  144  180  206  227  feet. 

The    practical   horizontal    distance    for    an    effective    fire- 
stream,  is,  however,  only  about  one-half  of  these  figures. 

The  ball  nozzle,  often  used  for  sprinkling,  has  a  cup  at 
the  end  of  the  nozzle  and  within  the  cup  a  ball,  so  that  the 
jet  issuing  from  the  tip  of  the  nozzle  is  deflected  side  wise 
in  all  directions.  This  apparatus  exhibits  a  striking  illus- 
tration of  the  principle  of  negative  pressure,  for  the  ball 
is  not  driven  away  from  the  tip,  but  is  held  close  to  it  by 
the  atmospheric  pressure,  the  negative  pressure-head  being 
caused  by  the  high  velocity  of  the  sheet  of  water  around  the 
ball.  The  cup  is  usually  so  arranged  that  the  ball  cannot 
be  driven  out  of  it,  for  this  might  occur  under  the  first 
impact  of  the  jet,  but  when  the  flow  has  become  steady 
there  is  no  tendency  of  this  kind,  and  the  ball  is  seen  slowly 
revolving  without  touching  any  part  of  the  cup. 

Prob.  80a.  Find  from  Table  32  the  heights  of  vertical  jets 
for  a  f-inch  and  a  i|-inch  nozzle,  and  the  discharges  in  gallons 
per  minute,  when  the  indicated  pressure  at  the  entrance  is  75 
pounds  per  square  inch. 

Prob.  806.  A  nozzle  if  inches  in  diameter  attached  to  a 
play-pipe  i\  inches  in  diameter  discharges  310.6  gallons  per 
minute  under  an  indicated  pressure  of  30  pounds  per  square 
inch.  Find  the  velocity  of  the  jet  and  the  coefficient  c±. 

Prob.  80c.  Insert  a  pin  through  the  center  of  a  piece  of 
cardboard  about  2  inches  in  diameter.  Put  the  pin  into  one  end 
of  a  straw  tube  and  blow  hard  into  the  other  end.  Explain  the 
phenomena  which  are  observed. 


194  FLOW  THROUGH  TUBES  CHAP,  vir 


ART.  81.     LOST  HEAD  IN  LONG  TUBES 

When  water  issues  from  an  orifice,  tube,  pipe,  or  nozzle 
with  the  velocity  v,  its  velocity-head  is  vz/2g  and  it  is 
only  this  part  of  the  total  effective  head  h  that  can  be 
utilized  for  the  production  of  work.  The  lost  head  then  is 


Now  if  c±  be  the  coefficient  of  velocity  for  the  section 
where_the  discharge  occurs,  the  velocity  v  is  given  by 
cl\/2ght  and  hence 


is  a  general  expression  for  the  lost  head  in  terms  of  the 
velocity-head.  For  the  standard  orifice  (Art.  45),  the 
mean  value  of  c±  is  0.98  and  for  an  orifice  perfectly  smooth 
cl  is  i.  oo,  hence  from 


are  the  losses  of  head  for  these  two  cases. 

For    the    standard     short     cylindrical    tube    (Art.  76) 
the  value  of  c1  is  about  0.82,  and  the  loss  of  head  is 

i  \  v2  v2 


n        9  •*•     I  ^  •  *-r  V 

V0.822          /  2g  Yy2g 

For  the  inward  projecting  cylindrical  tube  (Art.  78)  the 
value  of  cl  is  about  0.72,  and  hence  the  loss  of  head  is 

Accordingly  the  loss  of  head  for  the  inward  projecting 
tube  is  nearly  equal  to  the  velocity-head  of  the  issuing 
stream,  while  that  from  the  standard  tube  is  about  one- 
half  the  velocity-head. 

When   a  tube   is  longer  than   three   diameters   it   be- 
comes a  long  tube  or  a  pipe.     Here  the  loss  of  head  is 


ART.  81  LOST   HEAD    IN    LONG   TUBES  195 

much  greater  because  the  water  meets  with  frictional 
resistances  along  the  interior  surface,  and  the  longer  the 
pipe  the  greater  is  this  resistance  and  the  slower  is  the 
velocity.  The  formula  (81)!  gives  the  total  loss  of  head 
for  this  case  also.  For  example,  the  experiments  of 
Eytelwein  and  others  have  given  values  of  cl  for  the 
cases  below,  and  from  these  the  corresponding  values  of 
the  total  lost  head  have  been  computed.  If  /  denotes  the 
length  of  the  pipe  and  d  its  diameter,  the  end  connected 
with  the  reservoir  being  arranged  like  the  standard  tube; 
then 

<:i=:0-77  h'  =o.6c)V2/2g 

^  =  0.67  h'  =  i.23V2/2g 

^=0.60  h'  =  i.'jjv2/2g 

Now  in  each  of  these  cases  the  amount  o.4gv2/2g  is  lost 
in  entering  the  tube  and  in  impact,  as  in  the  standard 
short  tube.     Hence  the    loss  of  head  in   friction  in    the 
remaining  length  of  the  pipe  is  h"  =  h'  —  o.^v2/2g,  or 
for/  =  i2d  h"  =  0.20V2/  2g 

h"  = 


which  show  that  the  frictional  losses  increase  with  the 
length  of  the  pipe.  The  length  of  the  pipe  in  which 
the  entrance  losses  occur  is  about  $d\  hence  if  $d  be 
subtracted  from  each  of  the  above  lengths,  the  lengths 
in  which  the  friction  loss  occurs  are  gd,  33^,  and  570?, 
and  it  is  seen  that  the  above  losses  of  head  in  friction 
are  closely  proportional  to  these  lengths.  By  these  and 
many  other  experiments  it  has  been  shown  that  the  loss  of 
head  in  friction  varies  directly  with  the  length  of  the  pipe. 

The  lost  head  has  here  been  expressed  in  terms  of 
the  velocity-head,  but  it  can  also  be  expressed  in  terms 
of  the  total  head  h  that  causes  the  flow.  For,  substituting 
in  (81)!  the  value  of  v  given  by  c1\/2gh,  it  reduces  to 


196 


FLOW    THROUGH    TUBES 


CHAP.  VII 


Thus,  for  the  standard  short  tube  h'  =0.33/2;  for  the  in- 
ward projecting  tube  kf  =  0.48 h,  and  for  the  above  tube 
or  pipe  whose  length  is  60  diameters  h'  =0.64/2. 

Prob.  81a.  If  a  standard  orifice  and  a  standard  tube  be  of 
the  same  diameter,  show  that  the  former  will  deliver  about 
6  percent  more  power  than  the  latter. 


ART.  82.     INCLINED  TUBES  AND  PIPES 

The  tubes  discussed  in  this  chapter  have  generally 
been  regarded  as  horizontal,  but,  if  this  is  not  the  case, 
the  formulas  for  velocity  and  discharge  may  be  applied 
to  them  by  measuring  the  head  from  the  water  level  in 
the  reservoir  down  to  the  center  of  the  head  of  the  pipe. 
Thus,  for  the  nozzles  of  Art.  80,  it  is  understood  that  the 
tip  is  at  the  same  level  as  the  gage  which  registers  the 
pressure  pl  or  the  pressure-head  h1 ;  if  the  tip  be  lower  than 
the  gage  by  the  vertical  distance  dlt  the  true  pressure- 
head  to  be  used  in  the  formula  is  /^H-c^;  if  it  be  higher 
the  true  pressure-head  is  kv  —  d^  Then  the  velocity-head 
v2/2g  is  to  be  measured  upward  from  the  tip  of  the  nozzle. 

The  theorem  of  Bernouilli,  given  in  Art.  32,  is  true 
for  inclined  as  well  as  for  horizontal  pipes  under  uniform 
flow,  but  it  will  be  convenient  to  express  it  in  a  slightly 

different  form.  Let 
ax  and  a2  be  two 
sections  of  a  pipe, 
where  the  velocities 
are  ^  and  v2,  and 
the  pressure  -  heads 
are  h^  and  h2,  and 

let  the  flow  be  steady 
FIG.  82 

so     that     the     same 

weight  of  water,   w,   passes  each  section  in  one  second. 
Let  MN  be  any  horizontal  plane  lower  than  the  lowest 


ART.  82  INCLINED  TUBES  AND  PIPES  197 

section,  as  for  instance  the  sea  level,  and  let  el  and  e2  be 
the  elevations  of  a1  and  a2  above  it.  With  respect  to 
this  plane  the  weight  W  at  a^  has  the  potential  energy 
Wel  the  pressure-energy  Wh^  and  the  kinetic  energy 
VV.v12/2g,  or  the  total  energy  is 


Similarly  with  respect  to  this  plane  the  energy  in  a3  is 


If  no  losses  of  energy  occur  between  the  two  sections, 
these  expressions  are  equal,  and  hence 

«i  +  *i  +  ^-'a  +  *»  +  ^  (82), 

and  accordingly  the  theorem  may  be  stated  as  follows  : 

In  any  pipe,  under  steady  flow  without  impact  or  friction, 
the  gravity-head  plus  the  pressure-head  plus  the  velocity- 
head  is  a  constant  quantity  for  every  section. 

Now  let  Hl  and  H2  be  the  heights  of  the  water  levels  in 
the  piezometer  tubes  above  the  datum  plane;  then  e1  + 
hl=H1  and  e2+h2=H2)  and  accordingly  (82)  l  becomes 

H>+TrH>+V-fs  (82)> 

or,  the  piezometer  elevation  for  a^  plus  the  velocity-head 
is  equal  to  the  sum  of  the  corresponding  quantities  for 
any  other  section. 

This  theorem  belongs  to  theoretical  hydraulics,  in 
which  frictional  resistances  are  not  considered.  Under 
actual  conditions  there  is  always  a  loss  of  energy  or  head, 
so  that  when  water  flows  from  a1  to  a2  the  first  member 
of  the  above  equation  is  larger  than  the  second.  Let 
Wh'  be  the  loss  in  energy,  then  this  is  equal  to  the  dif- 


198  FLOW  THROUGH  TUBES  CHAP,  vii 

ference  of  the  energies  in  al  and  a2  with  respect  to  the 
datum  plane,  and 


or       '  h>=Hl-H,  +  fg-fg  (82). 

that  is,  the  lost  head  is  equal  to  the  difference  in  level 
of  the  water  surfaces  in  the  piezometer  tubes  plus  the 
difference  of  the  velocity-heads.  If  the  pipe  be  of  the 
same  size  at  the  two  sections,  the  velocities  vl  and  v2  are 
equal  when  the  flow  is  uniform,  and  the  lost  head  is  simply 

h'-H^H,  (82)4 

Piezometers  or  pressure  gages  hence  furnish  a  very  con- 
venient method  of  determining  the  head  lost  in  friction 
in  a  pipe  of  uniform  size.  For  a  pipe  of  varying  section 
the  velocities  v^  and  v2  must  also  be  known,  in  order  to 
use  (82)  3  for  finding  the  lost  head. 

Prob.  82.  A  large  Venturi  water  meter  placed  in  a  pipe  of 
57.823  square  feet  cross-section,  had  an  area  of  7.047  square 
feet  at  the  throat.  When  the  discharge  was  54.02  cubic  feet 
per  second,  the  elevations  of  the  water  levels  in  the  piezometers 
at  at  and  a2  in  Fig.  38a  were  99.858  and  98.951  feet.  Compute 
the  loss  of  head  between  the  two  sections. 


ART.  83.     VELOCITIES  IN  A  CROSS-SECTION 

Thus  far  the  velocity  has  been  regarded  as  uniform 
over  the  cross-section  of  the  tube  or  pipe.  On  account 
of  the  roughness  of  the  surface,  however,  the  velocity 
along  the  surface  is  always  smaller  than  that  near  the 
middle  of  the  cross-section.  There  appears  to  be  no 
theoretical  method  of  finding  the  law  which  connects  the 
velocity  of  a  filament  with  its  distance  from  the  center  of 
the  pipe,  and  yet  it  is  probable  that  such  a  law  exists. 


ART.  83  VELOCITIES    IN    A    CROSS-SECTION  199 

The  mean  velocity  is  evidently  greater  than  the  velocity 
at  the  surface  and  less  than  the  velocity  at  the  middle, 
and  if  the  position  of  a  filament  were  known  whose  ve- 
locity is  the  same  as  the  mean  velocity,  a  Pitot  tube 
(Art.  41)  with  its  tip  at  that  position  would  directly 
measure  the  mean  velocity. 

Let  Fig.  83  be   a  longitudinal  section  of  a  pipe,  and 
let  AB  be  laid  off  to  represent  the  surface  velocity  v8  and 
CD  to  represent  the  central  velocity 
•vc.     Then    the    velocity    v    at    any 
distance  y  from  the  axis  will  be  an 
abscissa    parallel    to    the    axis    and 
limited    by    the    line    AC    and    the 
curve   BD.     Suppose   this   curve   to 

be  a  parabola  whose  equation  is  y2=mx,  the  origin  being 
at  D  and  x  measured  toward  the  left.  When  y  is  equal 
to  the  radius  of  the  pipe  r,  the  value  of  x  is  vc  —  v8  and 
hence  m  =  rz/(vc  —  vs).  The  velocity  vy'  at  the  distance 
y  above  the  axis  is  vc  —  x,  and  accordingly 


It  thus  is  seen  that  the  velocity  at  any  distance  from  the 
axis  cannot  be  found  unless  the  surface  and  central  ve- 
locities are  known.  The  position  of  the  filament  having 
the  same  velocity  as  the  mean  velocity  v  can,  however, 
be  determined,  since  the  mean  velocity  is  the  mean  length 
of  the  solid  of  revolution  whose  section  is  shown  by  the 
broken  lines.  This  solid  consists  of  a  cylinder  •  having 
the  volume  nr*v8  and  a  paraboloid  having  the  volume 
%nr*(vc  —  v8),  and  the  sum  of  these  is  %xr2(vc  +  v8).  Divid- 
ing this  by  the  area  of  the  cross-section  gives  %(ve  +  v±) 
as  the  value  of  the  mean  velocity,  and  inserting  this  for 
•vy  in  the  above  equation  there  is  found  y  =  o.jir  for  the 
ordinate  of  a  filament  whose  velocity  is  the  same  as  mean 
velocity  v.  If  the  parabolic  curve  gives  the  true  Jaw  of 
variation  of  velocity,  a  Pitot  tube  with  its  tip  placed 


200  FLOW   THROUGH   TUBES  CHAP.  VII 

o.2()r  below  tne  top  of  the  pipe  would  measure  the  mean 
velocity  directly. 

The  first  measurements  of  velocities  of  filaments  were 
made  by  Freeman  in  1888  with  the  Pitot  tube.*  They 
were  on  jets  issuing  from  fire  nozzles  and  also  from  a 
ij-inch  tube  under  high  velocities.  For  smooth  nozzles 
the  velocities  were  practically  constant  for  a  distance  of 
o.6r  from  the  center,  and  then  rapidly  decreased,  and 
the  ratio  of  the  surface  velocity  to  the  central  velocity 
was  about  0.77.  For  the  pipe  the  velocities  decreased 
quickly  near  the  center  but  more  rapidly  toward  the 
surface.  The  velocity  curve  for  the  nozzle  lies  outside 
and  that  for  the  pipe  lies  within  the  parabolic  curve  rep- 
resented by  the  equation  (83)  t. 

Bazin  made  experiments  in  1893  on  jets  from  stand- 
ard orifices,  using  also  the  Pitot  tube.f  He  found  the 
velocities  near  the  center  to  be  smaller  than  others  within 
o.2r  of  the  surface.  Thus  if  vy  =  c\/2gh  the  following 
are  some  of  his  values  of  c  for  a  vertical  circular  and  a 
vertical  square  orifice,  h  being  the  head  on  the  center. 

r=+o.8        +0.6        -f-o.2          o.o        —  0.2         —0.6      —0.8 
c=      0.68         0.64         0.62       0.63          0.64         0.72       0.86 
c=     0.71         0.67         0.64       0.64         0.65         0.71       0.82 

These  are  for  velocities  in  the  plane  of  the  orifice  and  he 
found  similar  variations  for  a  section  of  the  jet  at  a  dis- 
tance from  the  orifice  of  about  one-half  its  diameter. 

Cole,  in  1897,  made  measurements  of  velocities  in 
pipes,  I  using  the  Pitot  tube  with  a  differential  gage 
(Art.  37).  For  pipes  4,  6,  and  12  inches  in  diameter  he 
found  the  ratio  of  the  mean  velocity  to  the  center  velocity 
to  range  from  0.91  to  i.oi,  while  for  a  1 6 -inch  pipe  he 

*  Transactions  American  Society  Civil  Engineers,    1889,  vol.  21,  p.  412. 
f  Experiments   on   the  Contraction  of  the  Liquid  Vein.      Trautwine's 
translation,  New  York,  1896. 
J  Transactions  American  Society  Civil  Engineers,   1902,  vol.  47,  p.  276. 


ART.  83  VELOCITIES   IN    A   CROSS-SECTION  201 

found  it  to  range  from  0.83  to  0.86.  His  velocity  curves 
show  that  the  surface  velocity  was  sixty  percent  or  more 
of  the  center  velocity. 

Williams,  Hubbell,  and  Fenkell,  in  1899,  made  numer- 
ous measurements  of  velocities  in  water  mains  with  the 
Pitot  tube,  and  arrived  at  the  conclusions  that  the  ratio 
of  the  mean  velocity  to  the  central  velocity  was  about 
0.84,  and  that  the  surface  velocity  was  about  one-half 
the  central  velocity.*  These  ratios  agree  with  an  ellipse 
better  than  with  a  parabola.  Let  the  curve  BD  in  Fig. 
83  be  an  ellipse  having  the  semi-axes  ED  and  BE,  the 
ellipse  being  tangent  to  the  pipe  surface  at  B.  As  before 
let  AB  represent  the  surface  velocity  va  and  CD  the  central 
velocity  vc;  then  ED  is  vc  —  v8  and  BE  is  the  radius  r. 
The  equation  of  the  ellipse  with  respect  to  E  as  an  origin  is 


in  which  x  is  measured  toward  the  right  and  y  upward. 
The  velocity  vy  at  any  distance  y  from  the  axis  CD  is 
v9  +  x,  and  accordingly 

vy=v.  +  (ve-v.)Vi-y*/r*  (83), 

Now  the  mean  velocity  is  the  mean  length  of  the  solid 
of  revolution  formed  by  the  cylinder  whose  volume  is 
xr2v8  and  the  semi-ellipsoid  whose  volume  is  $nr*(vc  —  va). 
The  volume  of  the  solid  is  hence  Kr2($vc  +  $v8)  and  the 
mean  velocity  is  $vc  +  %v8.  Inserting  this  for  vy  in  (83), 
there  is  found  y=o.*]$r  for  the  position  of  the  filament 
having  the  same  velocity  as  the  mean  velocity,  while 
the  parabola  gave  y  =  o.jir.  If  v8  be  one-half  of  vc,  the 
mean  velocity  under  the  elliptic  law  is  $vc  +  $v8=o.&3vt 
while  under  the  parabolic  law  it  is  %ve  +  %vs  =0.75^. 

Much  irregularity  is  observed  in  velocity  curves  plotted 
from  actual  measurements,  this  being  due   to   pulsations 

*  Transactions  American  Society  Civil  Engineers,  1902,  vol.  47,  p.  63. 


202  FLOW  THROUGH  TUBES  CHAP,  vii 

in  the  water  and  to  errors  of  observations.  The  above 
experiments  were  on  pipes  having  diameters  of  12,  16, 
30,  and  42  inches  and  under  velocities  ranging  from  0.5 
to  7 . 5  feet  per  second ;  and  they  are  a  very  valuable  addi- 
tion to  the  knowledge  of  this  subject.  The  conclusion 
that  Vg  is  one-half  of  vc  is,  however,  one  that  appears  to 
be  liable  to  some  doubt.  The  conclusion  that  the  mean 
velocity  v  is  about  0.84^  appears  well  established,  and  a 
Pitot  tube  with  its  tip  at  the  center  of  the  pipe  will  hence 
determine  a  fair  value  of  the  mean  velocity,  several  read- 
ings being  taken  in  order  to  eliminate  errors  of  observation. 

Prob.  83.  Let  vs  =  s  .and  vc  =  6  feet  per  second.  Plot  the 
parabola  from  (83)!  and  the  ellipse  from  (83)2. 

ART.  84.     COMPUTATIONS  IN  METRIC  MEASURES 

Nearly  all  the  formulas  of  the  chapter  are  rational 
and  may  be  used  in  all  systems  of  measures.  In  the 
metric  system  lengths  are  to  be  taken  in  meters,  areas  in 
square  meters,  velocities  in  meters  per  second,  discharges 
in  cubic  meters  per  second,  and  using  for  the  acceleration 
constants  the  values  given  in  Table  12. 

(Art.  80)  The  coefficients  of  discharge  and  velocity 
for  smooth  fire  nozzles  2.0,  2.5,  3.0,  and  3.5  centimeters 
in  diameter  are  0.983,  0.972,  0.973,  ano^  °-959  respectively. 
In  using  the  formula  (80)  2  the  values  of  d  and  ht  should 
be  taken  in  meters,  but  in  finding  the  ratio  D/d  the  values 
of  D  and  d  may  be  in  centimeters  or  any  other  convenient 
unit.  The  constant  g  being  9.80  meters  per  second,  the 
discharge  q  will  be  in  cubic  meters  per  second.  When  it  is 
desired  to  use  the  gage  reading  pt  in  kilograms  per  square 
centimeter  and  to  take  D  in  centimeters,  the  formula 


<?=:65-96c'jD\I-^(d/D)4 
may  be  used  for  finding  the  discharge  in  liters  per  minute. 


ART.  84          COMPUTATIONS  IN  METRIC  MEASURES  203 

Prob.  84a.  Compute  the  loss  of  head  which  occurs  when  a 
pipe,  discharging  18.5  cubic  meters  per  second,  suddenly  en- 
larges in  section  from  30  to  40  centimeters. 

Prob.  846.  Find  the  coefficient  of  discharge  for  a  tube  8 
centimeters  in  diameter  when  the  flow  under  a  head  of  4  meters 
is  18.37  cubic  meters  in  5  minutes  and  15  seconds. 

Prob.  84c.  Compute  the  discharge  from  a  smooth  nozzle 
2.5  centimeters  in  diameter,  attached  to  a  hose  7.5  centimeters 
in  diameter,  when  the  pressure  at  the  entrance  is  5.2  kilograms 
per  square  centimeter. 

Prob.  84 d.  For  a  pipe  of  uniform  diameter  the  piezometer 
heights  in  two  tubes  at  joints  one  kilometer  apart  are  693.143 
meters  above  sea  level  when  there  is  no  flow.  If  the  loss  in 
friction  during  the  flow  is  0.032  meters  per  linear  meter  of  pipe, 
and  the  upper  piezometer  level  stands  at  the  elevation  650.043 
meters,  what  should  be  the  elevation  of  the  lower  piezometer 
level? 


204  FLOW   THROUGH    PIPES  CHAP.  VIII 


CHAPTER  VIII 

FLOW   OF   WATER   THROUGH    PIPES 

ART.  85.     FUNDAMENTAL  IDEAS 

Pipes  made  of  clay  were  used  in  very  early  times  for 
conveying  water.  Pliny  says  that  they  were  two  digits 
(0.73  inches)  in  thickness,  that  the  joints  were  filled  with 
lime  macerated  in  oil,  and  that  a  slope  of  at  least  one- 
fourth  of  an  inch  in  a  hundred  feet  was  necessary  in  order 
to  insure  the  free  flow  of  water.*  The  Romans  also  used 
lead  pipes  for  conveying  water  from  their  aqueducts  to 
small  reservoirs  and  from  the  latter  to  their  houses.  Fron- 
tinus  gives  a  list  of  twenty-five  standard  sizes  of  pipes,! 
varying  in  diameter  from  0.9  to  9  inches,  which  were 
made  by  curving  a  sheet  of  lead  about  ten  feet  long  and 
soldering  the  longitudinal  joint.  The  Romans  had  con- 
fused ideas  of  the  laws  of  flow  in  pipes,  their  method  of 
water  measurement  being  by  the  area  of  cross-section,  with 
little  attention  to  the  head  or  pressure.  They  knew  that 
the  areas  of  circles  varied  as  the  squares  of  the  diameters, 
and  their  unit  of  water  measurement  was  the  quinaria, 
this  being  a  pipe  ij  digits  in  diameter;  then  the  denaria 
pipe,  which  had  a  diameter  of  z\  digits,  was  supposed 
to  deliver  4  quinarias  of  water. 

In  modern  times  lead  pipes  have  also  been  used  for 
house  service,  but  these  have  now  been  largely  superseded 
by  iron  ones.  For  the  mains  of  city  water  supplies  cast- 
iron  pipes  are  most  common,  and  since  1890  steel-riveted 

*  Natural  History,  book  31,  chapter  31,  line  5. 

fHerschel,  Water  Supply  of  the  City  of  Rome  (Boston,  1899),  p.  36. 


ART.  85  FUNDAMENTAL   IDEAS  205 

pipes  have  come  into  use  for  large  sizes.  Lap-welded 
wrought-iron  or  steel  pipes  are  used  in  some  cases  where 
the  pressure  is  very  high,  and  large  wooden  stave  pipes 
are  in  use  in  the  western  part  of  the  United  States. 

The  simplest  case  of  the  flow  of  water  through  a  pipe 
is  that  where  the  diameter  of  the  pipe  is  constant  and 
the  discharge  occurs  entirely  at  the  open  end.  This  case 
will  be  discussed  in  Arts.  86-95,  and  afterwards  will  be 
considered  the  cases  of  pipes  of  varying  diameter,  a 
pipe  with  a  nozzle  at  the  end,  and  pipes  with  branches. 
Most  of  the  principles  governing  the  simple  case  apply 
with  slight  modification  to  the  more  complex  ones.  Pipes 
used  in  engineering  practice  range  in  diameter  from  \  inch 
up  to  6  feet. 

The  phenomena  of  flow  for  this  common  case  are 
apparently  simple.  The  water  from  the  reservoir,  as 
it  enters  the  pipe,  meets  with  more  or  less  resistance  de- 


FIG.  85a  FIG.  856 

pending  upon  the  manner  of  connecting,  as  in  tubes  (Art. 
78).  Resistances  of  friction  and  cohesion  must  then  be 
overcome  along  the  interior  surface,  so  that  the  discharge 
at  the  end  is  much  smaller  than  in  the  tube  (Art.  81). 
When  the  flow  becomes  steady,  the  pipe  is  entirely 
filled  throughout  its  length;  and  hence  the  mean  velocity 
at  any  section  is  the  same  as  that  at  the  end,  since  the 
size  is  uniform.  This  velocity  is  found  to  decrease  as 
the  length  of  the  pipe  increases,  other  things  being  equal, 
and  becomes  very  small  for  great  lengths,  which  shows 
that  nearly  all  the  head  has  been  lost  in  overcoming  the 
resistances.  The  length  of  the  pipe  is  measured  along 


206  FLOW  THROUGH  PIPES  CHAP,  vni 

its  axis,  following  all  the  curves,  if  there  be  any.  The 
velocity  considered  is  the  mean  velocity,  which  is  equal 
to  the  discharge  divided  by  the  area  of  the  cross-section 
of  the  pipe.  The  actual  velocities  in  the  cross-section 
are  greater  than  this  mean  near  the  center  and  less  than 
it  near  the  interior  surface  of  the  pipe,  the  law  of  distribu- 
tion being  that  explained  in  Art.  83. 

The  object  of  the  discussion  of  flow  in  pipes  is  to  enable 
the  discharge  which  will  occur  under  given  conditions 
to  be  determined,  or  to  ascertain  the  proper  size  which 
a  pipe  should  have  in  order  to  deliver  a  given  discharge. 
The  subject  cannot,  however,  be  developed  with  the 
definiteness  which  characterizes  the  flow  from  orifices, 
and  weirs,  partly  because  the  condition  of  the  interior 
surface  of  the  pipe  greatly  modifies  the  discharge,  partly 
because  of  the  lack  of  experimental  data,  and  partly  on 
account  of  defective  theoretical  knowledge  regarding 
the  laws  of  flow.  In  orifices  and  weirs  errors  of  two 
or  three  percent  may  be  regarded  as  large  with  careful 
work ;  in  pipes  such  errors  are  common,  and  are  generally 
exceeded  in  most  practical  investigations.  It  fortunately 
happens,  however,  that  in  most  cases  of  the  design  of 
systems  of  pipes  errors  of  five  and  ten  percent  are  not 
important,  although  they  are  of  course  to  be  avoided 
if  possible,  or,  if  not  avoided,  they  should  occur  on  the 
side  of  safety. 

The  head  which  causes  the  flow  is  the  difference  in 
level  from  the  surface  of  the  water  in  the  reservoir  to 
the  center  of  the  end,  when  the  discharge  occurs  freely 
into  the  air  as  in  Fig.  85a.  If  h  be  this  head,  and  W  the 
weight  of  water  discharged  per  second,  the  theoretic 
potential  energy  per  second  is  Wh ;  and  if  v  be  the  actual 
mean  velocity  of  discharge  the  kinetic  energy  of  the  dis- 
charge is  W.v2/2g.  The  difference  between  these  is 
the  energy  which  has  been  transformed  into  heat  in  over- 


ART.  85  FUNDAMENTAL    IDEAS  207 

coming  the  resistances.  Thus  the  total  head  is  h,  the 
velocity-head  of  the  outflowing  stream  is  v*/2g,  and  the 
lost  head  is  h  —  v*/2g.  If  the  lower  end  of  the  pipe  is 
submerged,  as  in  Fig.  856,  the  head  h  is  the  difference 
in  elevation  between  the  two  water  levels. 

The  total  loss  of  head  in  a  straight  pipe  of  uniform 
size  consists  of  two  parts,  as  in  a  long  tube  (Art.  81). 
First,  there  is  a  loss  of  head  h'  due  to  entrance,  which  is 
the  same  as  in  a  short  cylindrical  tube,  and  secondly 
there  is  a  loss  of  head  h"  due  to  the  frictional  resistance 
of  the  interior  surface.  The  loss  of  head  at  entrance 
is  always  less  than  the  velocity-head  and  in  this  chapter 
it  will  be  expressed  by  the  formula 

h'=m—  (85). 

2g 

in  which  m  is  0.93  for  the  inward  projecting  pipe,  0.49 
for  the  standard  end,  and  o  for  a  perfect  mouthpiece, 
as  shown  in  Art.  81.  When  the  condition  of  the  end 
is  not  specified,  the  value  used  for  m  will  be  0.5,  which 
supposes  that  the  arrangement  is  like  the  standard  tube, 
or  nearly  so.  For  short  pipes,  however,  it  may  be  necessary 
to  consider  the  particular  condition  of  the  end,  and  then 
m  is  to  be  computed  from 

m  =  (i/c1)2-i  (85), 

in  which  the  coefficient  cl  is  to  be  selected  from  the  evi- 
dence presented  in  the  last  chapter. 

It  should  be  noted  that  the  loss  of  head  at  entrance 
is  very  small  for  long  pipes.  For  example,  it  is  proved 
by  actual  gagings  that  a  clean  cast-iron  pipe  10  ooo  feet 
long  and  i  foot  in  diameter  discharges  about  4^  cubic 
feet  per  second  under  a  head  of  100  feet'.  The  mean  velocity 
then  is,  if  q  be  the  discharge  and  a  the  area  of  the  cross- 
section, 

v  =  —  =—       —  =  15.41  feet  per  second, 
a     0.7854 


208  FLOW  THROUGH  PIPES  CHAP,  vm 

and  the  probable  loss  of  head  at  entrance  hence  is 
/*'=o.5Xo.oi555X5-4i2=o.23  feet, 

or  only  one-fourth  of  one  percent  of  the  total  head.  In 
this  case  the  effective  velocity-head  of  the  issuing  stream 
is  only  0.45  feet,  which  shows  that  the  total  loss  of  head 
is  99.55  feet,  of  which  99.32  feet  are  lost  in  friction. 

Prob.  85.  Under  a  head  of  20  feet  a  pipe  i  inch  in  diameter 
and  100  feet  long  discharges  15  gallons  per  minute.  Compute 
the  loss  of  head  at  entrance. 

ART.  86.     Loss  OF  HEAD  IN  FRICTION 

The  loss  of  head  due  to  the  resisting  friction  of  the 
interior  surface  of  a  pipe  is  usually  large,  and  in  long  pipes 
it  becomes  very  great,  so  that  the  discharge  is  only  a 
small  percentage  of  that  due  to  the  head.  Let  h  be  the 
total  head  on  the  end  of  the  pipe  where  the  discharge 
occurs,  vz/2g  the  velocity-head  of  the  issuing  stream, 
h'  the  head  lost  at  entrance  and  h"  the  head  lost  in  friction. 
Then  if  the  pipe  be  straight  and  of  uniform  size,  so  that 
no  other  losses  occur, 


Inserting  for  the  entrance  -head  h'  its  value  from  Art.  85, 
this  equation  becomes 


which  is  a  fundamental  formula  for  the  discussion  of  flow 
in  straight  pipes  of  uniform  size. 

The  head  lost  in  friction  may  be  determined  for  a 
particular  case  by  measuring  the  head  h,  the  area  a 
of  the  cross-section  of  the  pipe,  and  the  discharge  per 
second^.  Then  q  divided  by  a  gives  the  mean  velocity 


ART.  85  LOSS    OF    HEAD    IN    FRICTION  209 

v,  and  from  the  above  equation,  inserting  for  m  its  value 
from  (85)  2  ,  there  is  found 


which  serves  to  compute  h"  ',  the  value  of  cl  being  first 
selected  according  to  the  condition  of  the  end.  This 
method  is  not  a  good  one  for  short  pipes  because  of  the 
uncertainty  regarding  the  coefficient  ^  (Art.  81),  but 
for  long  pipes  it  gives  precise  results. 

Another  method,  and  the  one  most  generally  employed, 
is  by  the  use  of  piezometers  (Art.  82).  A  portion  of  the 
pipe  being  selected  which  is  free  from  sharp  curves,  two 
piezometer  tubes  are  inserted  into  which  the  water  rises, 
or  the  pressure-heads  are  measured  by  gages  (Art.  36). 
The  difference  of  level  of  the  water  surfaces  in  the  piezom- 
eter tubes  is  then  the  head  lost  in  the  pipe  between  them 
(Art.  82),  and  this  loss  is  caused  by  friction  alone  if  the 
pipe  be  straight  and  of  uniform  size. 

By  these  methods  many  observations  have  been  made 
upon  pipes  of  different  sizes  and  lengths  under  different 
velocities  of  flow,  and  the  discussion  of  these  has  enabled 
the  approximate  laws  to  be  deduced  which  govern  the 
loss  of  head  in  friction,  and  tables  to  be  prepared  for 
practical  use.  These  laws  are  : 

i  .  The  loss  of  head  in  friction  is  proportional  to  the  length 
of  the  pipe. 

2.  It  increases  with  the  roughnesses  of  the  interior  surface. 

3.  It  decreases  as  the  diameter  of  the  pipe  increases. 

4.  It  increases  nearly  as  the  square  of  the  velocity. 

5.  It  'is  independent  of  the  pressure  of  the  water. 

These  five  laws  may  be  expressed  by  the  formula 

/^=/4—  (86) 


210  FLOW   THROUGH   PIPES  CHAP.  VIII 

in  which  /  is  the  length  of  the  pipe,  d  its  diameter,  /  is. 
an  abstract  number  which  depends  upon  the  degree  of 
roughness  of  the  surface,  and  v2/2g  is  the  velocity-head 
due  to  the  mean  velocity  of  flow. 

This  formula  may  be  justified  by  reasonings  based 
on  the  assumption  that  what  has  been  called  the  loss  in 
friction  is  really  caused  by  impact  of  the  particles  of 
water  against  each  other.  Fig.  86  represents  a  pipe  with 

the  roughness  of  its  surface 
|E|  enormously  exaggerated  and  im- 
-j.  perfectly  shows  the  disturbances 

thereby  caused.  As  any  particle 
I^G.  ge  of  water  strikes  a  protuberance 

on  the  surface,  it  is  deflected 

and  its  velocity  diminished,  and  then  other  particles 
of  water  in  striking  against  it  also  undergo  a  diminution 
of  velocity.  Now  in  this  case  of  impact  the  resisting 
force  F  acting  over  each  square  unit  of  the  surface  is- 
to  be  regarded  as  varying  with  the  square  of  the  velocity 
(Arts.  29  and  74).  The  total  resisting  friction  for  a  pipe 
of  length  /  and  diameter  d  is  then  xdlF,  and  the  work 
lost  in  one  second  is  ndlFv.  Let  W  be  the  weight  of  water 
discharged  in  one  second,  then  Wh"  is  also  the  energy 
lost  in  one  second.  But  W  =  wq,  if  w  be  the  weight  of 
a  cubic  unit  of  water  and  q  the  discharge  per  second, 
and  the  value  of  q  is  \nd?v.  Then,  equating  the  two 
expressions  for  the  lost  energy,  and  replacing  F  by  Cv* 
where  C  is  a  constant,  there  results 

^=±'F  =  4C/    , 

wd         w  d 

Now  C  must  increase  with  the  roughness  of  the  surface 
and  hence  this  expression  is  the  same  in .  form  as  (86) 
and  it  agrees  with  the  five  laws  of  experience. 

The  values  of  h"  having  been  found  by  experiments,. 
in  the  manner  explained  above,  values  of  the  quantity 


ART.  86  LOSS   OF    HEAD    IN    FRICTION  211 

/  can  be  computed.  In  this  way  it  has  been  found  that 
/  varies  not  only  with  the  roughness  of  the  interior  surface 
of  the  pipe,  but  also  with  its  diameter,  and  with  the  ve- 
locity of  flow.  From  the  discussions  of  Fanning,  Smith, 
and  others,  the  mean  values  of  /  given  in  Table  33  have 
been  compiled,  which  are  applicable  to  clean  cast-iron  and 
wrought-iron  pipes,  either  smooth  or  coated  with  coal-tar, 
and  laid  with  close  joints. 

The  quantity  /  may  be  called  the  friction  factor,  and 
the  table  shows  that  its  value  ranges  from  0.05  to  o.oi 
for  new  clean  iron  pipes.  A  rough  mean  value,  often 
used  in  approximate  computations,  is 

Friction  factor  /  =  o.o2  ^^ 

It  is  seen  that  the  tabular  values  of  /  decrease  both  when 
the  diameter  and  when  the  velocity  increases,  and  that 
they  vary  most  rapidly  for  srnall  pipes  and  low  velocities. 
The  probable  error  of  a  tabular  value  of  /  is  about  one 
unit  in  the  third  decimal  place,  which  is  equivalent  to 
an  uncertainty  of  ten  percent  when  /  =  o.on,  and  to  five 
percent  when  /  =  0.021.  The  effect  of  this  is  to  render 
computed  values  of  h"  liable  to  the  same  uncertainties; 
but  the  effect  upon  computed  velocities  and  discharges 
is  much  less,  as  will  be  seen  in  Art.  89. 

To  determine,  therefore,  the  probable  loss  of  head 
in  friction,  the  velocity  v  must  be  known,  and  /  is  taken 
from  Table  33  for  the  given  diameter  of  pipes.  The 
formula  (86)  then  gives  the  probable  loss  of  head  in  friction. 
F...r  example,  let  /=  10000  feet,  d=  i  foot,  ^  =  5.41  feet  per 
second.  Then,  from  Table  33  the  factor  /  is  0.021,  and 

h"  =  0.02 1X^4^X0.455=  95. 5  feet, 

which  is  to  be  regarded  as  an  approximate  value,  liable 
to  an  uncertainty  of  five  percent. 

From  Table  33  and  formula  (86)  the  losses  of  head 
in  friction  for  100  feet  of  clean  cast-iron  pipe  have  been 


212  (  FLOW  THROUGH  PIPES  CHAP,  vin 

computed  for  different  values  of  d  and  /  and  are  given 
in  Table  35,  from  which  approximate  computations  may 
be  rapidly  made.  Thus,  for  the  above  data,  by  inter- 
polation in  Table  35,  there  is  found  0.952  feet  for  the 
loss  in  100  feet  of  pipe,  and  then  for  10  ooo  feet  the  loss 
of  head  is  95.2  feet. 

Prob.  86.  Determine  the  actual  loss  of  head  in  friction  from 
the  following  experiment:  /  =  6o  feet,  ^  =  8.33  feet,  d  =  0.08 7 8 
feet,  (7  =  0.03224  cubic  feet  per  second,  and  c  =  o.8.  Compute 
the  probable  loss  for  the  same  data  from  formula  (86)  and  also 
from  Table  35. 

ART.  87.     Loss  OF  HEAD  IN  CURVATURE 

Thus  far  the  pipe  has  been  regarded  as  straight,  so 
that  no  losses  of  head  occur  except  at  entrance  and  in 
friction.  But  when  the  pipe  is  laid  on  a  curve  the  water 
suffers  a  change  in  direction  whereby  an  increase  of  pressure 
is  produced  in  the  direction  of  the  radius  of  the  curve 
and  away  from  its  center  (Art.  147).  This  increase  in 
pressure  causes  eddying  motions  of  the  water,  from  which 
impact  results  and  energy  is  transformed  into  heat.  The 
total  loss  of  head  k"r  due  to  any  curve  evidently  increases 
with  its  length,  and  should  be  greater  for  a  small  pipe 
than  for  a  large  one.  Hence  the  loss  of  head  due  to  the 
curvature  of  a  pipe  may  be  written 

v"-**Tg  (87)' 

in  which  I  is  the  length  of  the  curve,  d  the  diameter  of 
the  pipe,  v  the  mean  velocity  of  flow,  and  /t  is  an  abstract 
number  called  the  curve  factor,  that  depends  upon  the 
ratio  of  the  radius  of  the  curve  to  the  diameter  of  the  pipe. 
Let  R  be  the  radius  of  the  circle  in  which  the  center  line  of 
the  pipe  is  laid.  Then,  if  R  is  infinity,  the  pipe  is  straight 
and  /t  =  o;  but  as  the  ratio  R/d  decreases  the  value  of 
increases. 


ART.  87  LOSS   OF   HEAD    IN   CURVATURE  213 

There  are  few  experiments  from  which  to  determine 
the  values  of  /r  Weisbach,  about  1850,  from  a  discussion 
of  his  own  experiments  and  those  of  Castel,  deduced  a 
formula  for  the  value  of  fj/d  for  curves  of  one-fourth 
of  a  circle,*  and  from  this  the  following  values  of  the  curve 
factor  /j  have  been  computed : 

for  R/d=   20         10          -5  3          2  1.5        i.o 

^=0.004    0.008    0.016    0.030    0.047    0.072    0.184 

These  values  of  /x  are  applicable  only  to  small  smooth 
iron  pipes  where  the  entire  curve  is  without  joints,  since 
most  of  the  pipes  on  which  the  experiments  were  made 
were  probably  of  this  kind. 

Freeman,  in  1889,  made  measurements  of  the  loss  of 
head  in  fire  hose  2.49  and  2.64  inches  in  diameter,  and 
the  curves  were  complete  circles  of  2,  3,  and  4  feet  radius.  | 
From  the  results  given  for  the  smaller  hose  the  following 
values  of  the  curve  factor  /x  have  been  found : 

iorR/d=    19.2         14.4          9.6 
^=0.0033     0.0034     0.0048 

while  for  the  larger  hose  the  values  are 

iorR/d=    16.2         13.6         8.1 
^=0.0036     0.0046     0.0045 

These  values  are  in  fair  agreement  with  those  given  above 
for  the  small  iron  pipes. 

Williams,  Hubbell,  and  Fenkell,  in  1898  and  1899, 
made  measurements  in  Detroit  on  cast-iron  water  mains 
having  curves  of  90  degrees.  From  their  results  for  a 
3o-inch  pipe  the  values  of  the  curve  factor  f±  have  been 

*  Die  Experimental  Hydraulik  (Freiberg,  1855),  p.  159.     Mechanics  of 

Engineering  (New  York,  1870),  vol.  i,  p.  898. 

f  Transactions  American  Society  Civil  Engineers,   1889,  vol.  21,  p.  363. 


214  FLOW  THROUGH  PIPES  CHAP,  vin 

computed  and  are  found  to  be  as  follows  : 

for  R/d  =   24  16  10  6  4  2.4 

^=0.036      0.037      0.047      0.060      0.062      0.072 

while  from  their  work  on  a  12  -inch  pipe  the  values  are 

for  R/d  =4  3  2  i 

^=0.05      0.06      0.06      0.20 

Of  these  values,  those  derived  from  the  larger  pipe  are 
the  most  reliable,  and  it  is  seen  that  they  are  much  greater 
than  the  values  deduced  from  Weisbach's  investigations 
on  small  pipes.  Probably  some  of  this  increase  is  due 
to  the  circumstance  that  the  curves  had  rougher  surfaces 
and  that  the  joints  were  nearer  together  than  on  the 
straight  portions.  These  experiments*  were  made  with 
the  Pitot  tube  in  the  manner  explained  in  Arts.  41  and 
83.  They  show  that  the  law  of  distribution  of  the  ve- 
locities in  the  cross-section  is  quite  different  from  that 
for  a  straight  pipe,  the  maximum  velocity  being  not  at  the 
center  but  between  the  center  and  the  outside  of  the  curve. 

While  the  above  values  of  f1  are  few  in  number  they 
may  serve  as  a  basis  for  roughly  estimating  the  loss  of 
head  due  to  curvature.  For  example,  let  there  be  two 
curves  of  24  and  16  feet  radius  in  a  pipe  2  feet  in  diameter, 
each  curve  being  a  quadrant  of  a  circle.  The  ratios  R/d 
are  12  and  8,  and  the  values  of  /x,  taken  from  those  de- 
duced above  from  the  large  Detroit  pipe,  are  0.044  and 
0.053.  The  lengths  of  the  curves  are  37.7  and  25.1  feet, 
and  then  from  (87)  l 


*  Transactions  American  Society  of  Civil   Engineers,  1902,  vol.  47,  pp. 
185,  360. 


ART.  87  LOSS   OF   HEAD    IN    CURVATURE  215 

are  the  losses  of  head  for  the  two  cases.  Here  it  is  seen 
that  the  easier  curve  gives  the  greater  loss  of  head.  By 
the  use  of  the  values  of  fl  deduced  from  Weisbach's  in- 
vestigation, the  loss  of  head  is  much  smaller  and  the  sharper 
curve  gives  the  greater  loss  of  head,  since  the  coefficients 
of  the  velocity-head  are  found  to  be  0.13  and  0.14  instead 
of  0.83  and  0.66.  The  subject  of  losses  in  curves  is,  indeed, 
in  an  uncertain  state,  since  sufficient  experiments  have 
not  been  made  either  to  definitely  establish  the  validity 
of  (87)  l  or  to  determine  authoritative,  values  of  the  curve 
factor  /r  Probably  it  will  be  found  that  fl  varies  with 
the  diameter  d  as  well  as  with  the  ratio  R/d. 

When  there  are  several  curves  in  a  pipe  line  the  value 
of  iv(l/d)  for  each  curve  is  to  be  found  and  then  these 
are  to  be  added  in  order  to  find  the  total  loss  of  head. 
Thus,  in  general,  there  may  be  written, 

*'"-«,^  (87), 

for  the  total  loss  of  head,  in  which  ml  represents  the  sum 
of  the  values  of  f^l/d)  for  all  the  curves. 

The  lost  head  due  to  curvature  in  a  pipe  line  is  usually 
low  compared  with  that  lost  in  friction,  since  the  number 
of  curves  is  always  made  as  small  as  possible.  For  ex- 
ample, take  a  pipe  1000  feet  long  and  3  inches  in  diameter, 
which  has  ten  curves,  five  being  of  90  degrees  and  6  inches 
radius  and  five  being  of  57.3  degrees  and  5  feet  radius. 
From  (86),  using  0.02  for  the  mean  friction  factor,  the 
loss  of  head  in  friction  is  Sov2/2g.  From  (87)  lf  using  the 
curve  factors  deduced  from  Weisbach,  the  loss  of  head  for 
the  five  sharp  curves  is  o.j$v2/2g,  and  that  for  the  five  easy 
curves  is  o.4V2/2g. 

Prob.  87a.  Compute  the  values  in  the  last  sentence. 
Prob.   876.  If  the  central  angle  of  a  curve   of  63.5   inches 
radius  is  229°  06',  what  is  the  length  of  the  curve?     If  a  hose, 


216  FLOW   THROUGH    PIPES  CHAP.  VIII 

2\  inches  in  diameter,  be  laid  on  this  curve,  compute  the  loss  in 
head  due  to  curvature  when  the  velocity  in  the  hose  is  30  feet 
per  second  and  also  when  it  is  15  feet  per  second. 


ART.  88.     OTHER  LOSSES  OF  HEAD 

Thus  far  the  cross-section  of  the  pipe  has  been  supposed 
to  be  constant,  so  that  no  losses  of  head  occur  except 
at  entrance  (Art.  85),  in  friction  (Art.  86),  and  in  curvature 
(Art.  87).  But  if  the  pipe  contain  valves,  or  have  ob- 
structions in  its  cross-section,  or  be  of  different  diameters, 
other  losses  occur  which  are  now  to  be  considered. 

The  figures  show  three  kinds  of  valves  for  regulating 
the  flow  in  pipes :  A  being  a  valve  consisting  of  a  vertical 
sliding-gate,  B  a  cock-valve  formed  by  two  rotating 


FIG.  88 

segments,  and  C  a  throttle-  valve  or  circular  disk  which 
moves  like  a  damper  in  a  stovepipe.  The  loss  of  head 
due  to  these  may  be  very  large  when  they  are  sufficiently 
closed  so  as  to  cause  a  sudden  change  in  velocity.  It 
may  be  expressed  by 


in  which  m  has  the  following  values,  as  determined  by 
Weisbach  from  his  experiments  on  pipes  of  small  diameter.* 
For  the  gate-valve  let  d'  be  the  vertical  distance  that 
the  gate  is  lowered  below  the  top  of  the  pipe  ;  then 

for  (*'/<*  =o          i          i          !         i        I        I       1 
m=o.o     0.07     0.26     0.81     2.1     5.5      17     98 

*  Mechanics  of  Engineering,  vol.  i,  Coxe's  translation,  p.  902. 


ART.  88  OTHER    LOSSES    OF   HEAD  217 

For  the  cock- valve  let  6  be  the  angle  through  which  it.  is 
turned,  as  shown  at  B  in  Fig.  88 ;  then 

for  l9=o0     10°     20°     30°     40°  50°   55°     60°      65° 
m=o     0.29     1.6     5.5     17     53     106     206     486 

In  like  manner,  for  the  throttle- valve  the  coefficients  are : 

for  6=   5°        10°     20°     30°  40°    50°    60°     65°     70° 
w  =  o.24     0.52     1.5     3.9     ii     33     118     256     750 

The  number  m  hence  rapidly  increases  and  becomes  very 
great  when  the  valve  is  fully  closed,  but  as  the  velocity 
is  then  zero  there  is  no  loss  of  head.  The  velocity  v  here, 
as  in  other  cases,  refers  to  that  in  the  main  part  of  the 
pipe,  and  not  to  that  in  the  contracted  section  formed 
by  the  valve. 

Kuichling's  experiments*  on  a  gate-valve  for  a  24-inch 
pipe  give  values  of  m  which  are  somewhat  greater  than 
those  deduced  by  Weisbach  from  pipes  less  than  2  inches 
in  diameter.  Considering  the  great  variation  in  size 
the  agreement  is,  however,  a  remarkable  one.  He  found 

for  <*'/<* -i         T5*        i         I          f  ft 

w=o.8       1.6       3.3       8.6       22.7       41.2 

and  his  computed  value  of  m  for  the  case  where  d' /d 
equals  t  is  75-6- 

An  accidental  obstruction  in  a  pipe  may  be  regarded 
as  causing  a  sudden  expansion  of  section,  and  the  loss 
of  head  due  to  it  is,  by  Art.  74, 

inn       I*          YV*  V* 

h      =—7—1    —  =  m  — 

\a'       /  2g          2g 

where  a  is  the  area  of  the  section  of  the  pipe,  and  a'  that 
of  the  diminished  section.  This  formula  shows  that 
when  a'  is  one-half  of  a,  the  loss  of  head  is  equal  to  the 
velocity-head,  and  that  m  rapidly  increases  as  a'  diminishes. 

*  Transactions  American  Society  Civil  Engineers,  1892,  vol.  26,  p.  449. 


218  FLOW  THROUGH   PIPES  CHAP.  VIII 

The  same  formula  gives  the  loss  of  head  due  to  the  sudden 
enlargement  of  a  pipe  from  the  area  a'  to  a. 

Air-valves  are  placed  at  high  points  on  a  pipe  line 
in  order  to  allow  the  escape  of  air  that  collects  there. 
Mud-valves  or  blow-offs  are  placed  at  low  points  in  order 
to  clean  out  deposits  that  may  be  formed.  These  are 
arranged  so  as  not  to  ^contract  the  section  and  the  losses 
of  head  caused  by  them  are  generally  very  small.  When 
a  blow-off  pipe  is  opened  and  the  water  flows  through 
it  with  the  velocity  v,  the  loss  of  head  at  its  entrance, 
even  when  the  edges  are  rounded,  is  as  high  as  or  higher 
than  0.56  v2/2g,  according  to  the  experiments  of  Fletcher. 

In  the  following  pages  the  symbol  h""  will  be  used 
to  denote  the  sum  of  all  the  losses  of  head  due  to  valves 
and  contractions  of  section.  Then 

/j""=w—  (88) 

2  2g 

in  which  m2  will  denote  the  sum  of  all  the  values  of  m 
due  to  these  causes.  In  case  no  mention  is  made  regard- 
ing these  sources  of  loss  they  are  supposed  not  to  exist, 
so  that  both  m2  and  h""  are  simply  zero. 

Prob.  88.  Which  causes  the  greater  loss  in  a  24-inch  pipe, 
a  gate  valve  one-half  closed,  or  five  9o-degree  curves  of  16  feet 
radius  ? 

ART.  89.     FORMULA  FOR  MEAN  VELOCITY 

The  mean  velocity  in  a  pipe  can  now  be  deduced  for 
the  condition  of  steady  flow.  The  total  head  being  //, 
and  the  effective  velocity-head  of  the  issuing  stream 
being  v*/2g,  the  lost  head  is  h  —  v2/2g,  and  this  must  be 
equal  to  the  sum  of  its  parts,  or 

h-—  = 


ART.  89  FORMULA    FOR   MEAN   VELOCITY  219 

Substituting  in  this  the   values   9f  the   four  lost  heads, 
as  determined  in  the  four  preceding  articles,  it  becomes 


h  ---  =  m  —  +  /-j  , 

2g  2g       'dig  I2g 

and  by  solving  for  v  there  is  found 


=\ 


(89)' 


which  is  the   general  formula  for  the  mean  velocity  in 
a  pipe  of  constant  cross-section. 

The  most  common  case  is  that  of  a  pipe  which  has 
no  curves,  or  curves  of  such  large  radius  that  their  influ- 
ence is  very  small,  and  which  has  no  partially  closed  valves 
or  other  obstructions.  For  this  case  both  ml  and  m2 
are  zero,  and,  taking  m  as  0.5,  the  formula  becomes 

-^wm  (89)* 

which  applies  to  the  great  majority  of  cases  in  engineering 
practice. 

In  this  formula  the  friction  factor  /  is  a  function 
of  v  to  be  taken  from  Table  33,  and  hence  v  cannot  be 
directly  computed,  but  must  be  obtained  by  successive 
approximations.  For  example,  let  it  be  required  to  com- 
pute the  velocity  of  discharge  from  a  pipe  3000  feet  long 
and  6  inches  in  diameter  under  a  head  of  9  feet.  Here 
/  =  3ooo,  ^=0.5,  and  h  =  g  feet,  and  taking  for  /  the  rough 
mean  value  0.02,  formula  (89) 2  gives 

2X32.16X9 

2.2  feet  per  second. 


:. 5+0. 02X3000X2 
The  approximate  velocity  is  hence  2.2  feet  per  second, 
and  entering  the  table  with  this,  the  value  of  /  is  found 
to  be  0.026.  Then  the  formula  gives 

2X32.16X0 

-  =  1.92  feet  per  second. 
1.5+0.026X3000X2 


220  FLOW  THROUGH  PIPES  CHAP,  vin 

This  is  to  be  regarded  as  the  probable  value  of  the  ve- 
locity, since  the  table  gives  /  =  0.026  for  v  =  1.92.  In  this 
manner  by  one  or  two  trials  the  value  of  v  can  be  com- 
puted so  as  to  agree  with  the  corresponding  value  of  /. 

The  error  in  the  computed  velocity  due  to  an  error 
of  one  unit  in  the  last  decimal  of  the  friction  factor  /  is 
always  relatively  less  than  the  error  in  /  itself.  For  in- 
stance, if  v  be  computed  for  the  above  example  with  /  =  0.02  5 , 
which  is  four  percent  less  than  0.026,  its  value  is  found 
to  be  1.96  feet  per  second,  or  two  percent  greater  than 
1.92.  In  general  the  percentage  of  error  in  v  is  less  than 
one-half  of  that  in  /.  It  hence  appears  that  computed 
velocities  are  liable  to  probable  errors  ranging  from  one 
to  five  percent,  owing  to  imperfections  in  the  tabular 
values  of  /,  for  riew  clean  pipes.  This  uncertainty  is  as 
a  rule  still  further  increased  by  various  causes,  so  that 
five  percent  is  to  be  regarded  as  a  common  probable  error 
in  computations  of  velocity  and  discharge  from  pipes. 

Velocities  greater  than  15  feet  per  second  are  very 
unusual  in  pipes,  and  but  little  is  known  as  to  the  values 
of  /  for  such  cases.  For  velocities  less  than  0.5  feet  per 
second,  the  values  of  /  are  also  not  known  (Art.  103), 
so  that  only. a  rough  reliance  can  be  placed  upon  com- 
putations. The  usual  velocity  in  water  mains  is  less 
than  five  feet  per  second,  it  being  found  inadvisable  to 
allow  swifter  flow  on  account  of  the  great  loss  of  head 
in  friction. 

To  illustrate  the  use  of  the  general  formula  (89)  l  let 
the  pipe  in  the  above  example  be  supposed  to  have  forty 
9o-degree  curves  of  6  inches  radius,  and  to  contain  two 
gate-valves  which  are  half  closed.  Then  from  Arts.  87 
and  88  there  are  found  ml  =  n.6  for  the  curves  and  m2  =4.2 
for  the  gates.  The  mean  velocity  then  is 

I       2X32.16X9 

v**\-  -=1.83  feet  per  second, 

>f  17. 3 +0.026X6000 


ART.  90  COMPUTATION  OF  DISCHARGE  221 

which  is  but  a  trifle  less  than  that  found  before.  With 
a  shorter  pipe,  however,  the  influence  of  the  curves  and 
gates  in  retarding  the  flow  would  be  more  marked. 

The  head  required  to  produce  a  given  velocity  v  can 
be  obtained  from  (89^  or  (89)  r  Thus  from  the  general 
formula  the  required  head  is 


\  v* 
-- 

/       o 


in    which    for    common    computations    7^  =  0.5,    while    ml 
and  m2  are  neglected. 

Prob.  89a.  Compute  the  mean  velocities  for  the  above  ex- 
amples if  the  pipe  be  1000  feet  long. 

Prob.  896.  Using  for  /  the  mean  value  0.02,  compute  the 
head  required  to  cause  a  velocity  of  10  feet  per  second  in  a  pipe 
15  ooo  feet  long  and  1.5  feet  in  diameter. 

ART.  90.     COMPUTATION  OF  DISCHARGE 

The  discharge  per  second  from  a  pipe  of  given  diameter 
is  found  by  multiplying  the  velocity  of  discharge  by  the 
area  of  the  cross-section  of  the  pipe,  or 

q=lxd2v=o.'j&$4d'*v  (90) 

i 
in  which  v  is  to  be  found  by  the  method  of  the  last  article. 

For  example,  let  it  be  required  to  find  the  discharge 
in  gallons  per  minute  from  a  clean  pipe  3  inches  in  diameter 
and  1500  feet  long  tinder  a  head  of  64  feet.  Here  ^=0.25, 
/  =  1500,  and  h  =64  feet.  Then  for  /  =  o.o2  the  velocity 
is  found  from  (89) 2  to  be  5.82  feet  per  second;  then  from 
Table  33  is  found  /  =  o.o24  and  the  velocity  is  5.30  feet 
per  second.  The  discharge  in  cubic  feet  per  second  is 

q  =0.7854  Xo.252X  5-30  =0.260 

which  is  equal  to   116.7  gallons  per  minute.     This  is  the 
probable  result,  which  is  liable  to  the  same  uncertainty 


222  FLOW  THROUGH   PIPES  CHAP.  VIIL 

as  the  velocity,  say  about  three  percent;  so  that  strictly 
the  discharge  should  be  written  116.7  ±3. 6  gallons  per 
minute. 

By  inserting  the  value  of  v  from  (89)  a  in  the  above 
expression  for  q  it  becomes 


2gh 


:.5 +#*/<*) 

and  from  this  the  value  of  the  head  required  to  produce 
a  given  discharge  is 


These  formulas  are  not  more  convenient  for  precise  computa- 
tions than  the  separate  expressions  for  v,  q,  and  h  previously 
established,  since  v  must  be  computed  in  order  to  select 
/  from  the  table.  For  approximate  computations,  how- 
ever, when  /  may  be  taken  as  0.02,  they  may  advantageously 
be  used.  In  the  English  system  of  measures  h  and  d 
are  to  be  taken  in  feet  and  q  in  cubic  feet  per  second, 
and  the  constants  in  these  two  formulas  have  the  values 


7rg  =  0.0252 

The  last  formula  shows  that  the  head  required  for  a  pipe  of 
given  diameter  varies  directly  as  the  square  of  the  proposed 
discharge.  Thus,  if  a  head  of  50  feet  delivers  8  cubic 
feet  per  second  through  a  certain  pipe,  a  head  of  about 
200  feet  will  be  necessary  in  order  to  obtain  16  cubic  feet 
per  second. 

Prob.  90a.  Compute  the  probable  discharge  from  a  pipe  i 
inch  in  diameter  and  1000  feet  long  under  a  head  of  10  feet. 

Prob.  906.  What  head  is  required  to  discharge  6  gallons  per 
minute  through  a  pipe  i  inch  in  diameter  and  1000  feet  long? 

Prob.  90c.  Compute  the  probable  discharge  from  a  pipe  4 
inches  in  diameter  and  630  feet  long  under  a  head  of  25,  when 
it  has  three  Qo-degree  curves  of  4  feet  radius. 


ART.  91  COMPUTATION  OF  DIAMETER  223 

ART.  91.     COMPUTATION  OF  DIAMETER 

It  is  an  important  practical  problem  to  determine 
the  diameter  of  a  pipe  to  discharge  a  given  quantity  of 
water  under  a  given  head  and  length.  The  last  equation 
above  serves  to  solve  this  case,  if  the  curve  and  valve 
resistances  be  omitted,  as  all  the  quantities  in  it  except 
d  are  known.  This  equation  reduces  to  the  form 


and  for  the  English  system  of  measures  this  becomes 

(91) 


which  is  the  formula  for  computing  d  when  h,  /,  and  d 
are  in  feet  and  q  is  in  cubic  feet  per  second.  The  value 
of  the  friction  factor  /  may  be  taken  as  0.02  in  the  first 
instance,  and  the  d  in  the  right-hand  member  being 
neglected,  an  approximate  value  of  the  diameter  is  com- 
puted.  The  velocity  is  next  found  by  the  formula 

v  =  q/a  =  g/o.  78540^ 

and  from  Table  33  the  value  of  /  thereto  corresponding 
is  selected.  The  computation  for  d  is  then  repeated, 
placing  in  the  right-hand  member  the  approximate  value 
of  d.  Thus  by  one  or  two  trials  the  diameter  is  computed 
which  will  satisfy  the  given  conditions. 

For  example,  let  it  be  required  to  determine  the  diam- 
eter of  a  new  pipe  which  will  deliver  500  gallons  per  second, 
its  length  being  4500  feet  and  the  head  24  feet.  Here  the 
discharge  is 

q  =  500/7.  481  =66.84  cubic  feet  per  second. 
The  approximate  value  of  d  then  is 


/o.o2X45ooX66.842\i  .  , 

0.479^—          —  ^—          -  )    =3.35  feet. 


224  FLOW  THROUGH  PIPES  CHAP,  vin 

From  this  the  mean  Velocity  of  flow  is 
66.84 


o.  7854X3.  35' 

and  from  the  table  the  value  of  /  for  this  diameter  and 
velocity  is  found  to  be  0.013.  Then 

r/  ,66.842-^ 

d  =  o.  479^(1.5X3-  35+o.  013  X45oo)—  ——  J 

from  which  ^  =  3.125  feet.  With  this  value  of  d  the 
velocity  is  now  found  to  be  8.71  feet,  so  that  no  change 
results  in  the  value  of  /.  The  required  diameter  of  the 
pipe  is  therefore  3.1  feet,  or  about  37  inches;  but  as  the 
regular  market  sizes  of  pipes  furnish  only  36  inches  and 
40  inches,  one  of  these  must  be  used,  and  it  will  be  on 
the  side  of  safety  to  select  the  larger. 

It  will  be  well  in  determining  the  size  of  a  pipe  to  also 
consider  that  the  interior  surface  may  become  rough 
by  erosion  and  incrustation,  thus  increasing  the  value 
of  the  friction  factor  and  diminishing  the  discharge.  It 
has  been  found  that  some  waters  deposit  incrustations 
which  in  a  few  years  render  the  values  of  /  more  than 
double  those  given  in  Table  33.  The  increase  in  /  from 
these  causes  is  not  likely  to  be  so  great  in  a  large  pipe 
as  in  a  small  one,  but  it  is  not  improbable  that  for  the  above 
example  they  might  be  sufficient  to  make  /  as  large  as 
0.03.  Applying  this  value  to  the  computation  of  the 
diameter  from  the  given  data  there  is  found  ^  =  3.6  feet  = 
about  43  inches. 

The  sizes  of  iron  pipes  generally  found  in  the  market 
are  J,  },  i,  i^,  if,  2,  3,  4,  6,  8,  10,  12,  16,  18,  20,  24,  27, 
30,  36,  40,  44,  and  48  inches,  while  intermediate  and 
larger  sizes  must  be  made  to  order.  The  computation 
of  the  diameter  is  merely  a  guide  to  enable  one  of  these 
sizes  to  be  selected,  and  therefore  it  is  entirely  unnecessary 
that  the  numerical  work  should  be  carried  to  a  high  degree 


ART.  92  SHORT  PlPES  225 

of  precision.  In  fact,  three-figure  logarithms  are  usually 
sufficient  to  determine  reliable  values  of  d  from  formula 
(91). 

Prob.  91.  Compute  the  diameter  of  a  pipe  to  deliver  50 
gallons  per  minute  under  a  head  of  4  feet  when  its  length  is 
500  feet.  Also  when  its  length  is  5000  feet. 

ART.  92.     SHORT  PIPES 

A  pipe  is  said  to  be  short  when  its  length  is  less  than 
about  500  times  its  diameter,  and  very  short  when  the 
length  is  less  than  about  50  diameters.  In  both  cases 
the  coefficient  c1  should  be  estimated  according  to  the 
condition  of  the  upper  end  as  precisely  as  possible,  and 
the  length  /  should  not  include  the  first  three  diameters 
of  the  pipe,  as  that  portion  properly  belongs  to  the  tube 
which  is  regarded  as  discharging  into  the  pipe.  In  attempt- 
ing to  compute  the  discharge  for  such  pipes,  it  is  often 
found  that  the  velocity  is  greater  than  given  in  Table  33, 
and  hence  that  the  friction  factor  /  cannot  be  ascertained. 
For  this  reason  no  accurate  estimate  can  be  made  of  the 
discharge  from  short  pipes  under  high  heads,  and  fortu- 
nately it  is  not  often  necessary  to  use  them  in  engineering 
constructions. 

For  example,  let  it  be  required  to  compute  the  velocity 
of  flow  from  a  pipe  i  foot  in  diameter  and  100  feet  long 
under  a  head  of  100  feet,  the  upper  end  being  so  arranged 
that  ^=0.80,  and  hence  7^  =  0.56  (Art.  85.)  Neglecting 
ml  and  mv  since  the  pipe  has  no  curves  or  valves,  formula 
(89)  1  for  the  velocity  becomes 


2gh 


.56+f(l/d) 

and,  using  for  /  the  rough  mean  value  0.02  and  taking  Z 
as  97  feet,  there  is  found  42.9  feet  per  second  for  the  mean 
velocity.  Now  there  is  no  experimental  knowledge  regard- 


226  FLOW  THROUGH  PIPES  CHAP,  vin 

ing  the  value  of  the  friction  factor  /  for  such  high  velocities 
in  iron  pipes,  but  judging  from  the  table  it  is  probable 
that  /  may  be  about  0.015.  Using  this  instead  of  0.02 
gives  for  v  the  value  46  feet  per  second.  The  uncertainty 
of  this  result  should  be  regarded  as  at  least  ten  percent. 

The  general  equation  for  the  velocity  of  discharge 
deduced  in  Art.  89  may  be  applied  to  very  short  pipes 
by  writing  /  —  $d  in  place  of  /,  and  placing  for  m  its  value 
in  terms  of  the  coefficient  cr  It  then  becomes 


(92) 

If  in  this  /  equals  3^,  the  velocity  is  ^V^gS,  which  is  the 
same  as  for  the  short  cylindrical  tube.  If  I  =  i2d,  /  =  o.o2 
and  ^=0.82,  it  gives  v=  o.fj4\/2ght  which  agrees  well 
with  the  value  given  in  Art.  81  for  this  case.  If  /  =  6od, 
it  gives  v  =  o.6is\/2gh,  which  is  two  percent  greater  than 
the  value  given  in  Art.  81. 

Prob.  92.  Compute  the  discharge  per  second  for  a  pipe 
i  inch  in  diameter  and  40  inches  long  under  a  head  of  4  feet. 

ART.  93.     LONG  PIPES 

For  long  pipes  the  loss  of  head  at  entrance  becomes  very 
small  compared  with  that  lost  in  friction,  and  the  velocity- 
head  is  also  small.  Formula  (89)  2  for  the  mean  velocity  is 


2gh 


:-S +/('/« 

in  which  the  first  term  in  the  denominator  represents  the 
effect  of  the  velocity-head  and  the  entrance-head,  the 
mean  value  of  the  latter  being  0.5.  Now  it  may  safely 
be  assumed  that  1.5  may  be  neglected  in  comparison 
with  the  other  term,  when  the  error  thus  produced  in 


ART.  93  LONG   PlPES  227 

v  is  less  than  one  percent.     Taking  for  /  its  mean  value, 
this  will  be  the  case  when 


.         . 

==£—  =  i.oi,     whence    -3  =  375° 

V0.02//d  « 

Therefore  when  /  is  greater  than  about  4000^  the  pipe 
will  be  called  long. 

For  long  pipes  under  uniform  flow  the  velocity  is 
found  from  the  above  equation  by  dropping  1.5,  and  the 
discharge  is  found  by  multiplying  this  mean  velocity  by 
the  area  of  the  cross-section.  Hence 


which  for  the  English  system  of  measures  becomes 

Ihd  hd* 


(93), 

From  these  expressions  for  q  the  general  and  special  for- 
mulas for  computing  the  diameter  of  the  pipe  for  a  given 
discharge,  length,  and  head  are  found  to  be 


These  equations  show  that  for  very  long  pipes  the  dis- 
charge varies  directly  as  the  2\  power  of  the  diameter, 
and  inversely  as  the  square  root  of  the  length. 

In  the  above  formulas  d,  h,  and  I  are  to  be  taken  in 
feet,  q  in  cubic  feet  per  second,  and  /  is  to  be  found  from 
Table  33,  an  approximate  value  of  v  being  first  obtained 
by  taking  /  as  0.02.  It  should  not  be  forgotten  that 
computations  of  discharge  or  diameter  from  these  formulas 
are  liable  to  uncertainty  on  account  of  imperfect  knowl- 
edge regarding  the  friction  factors.  Especially  when  the 
velocities  are  lower  than  one  or  higher  than  fifteen  feet 


228  FLOW  THROUGH  PIPES  CHAP,  vin 

per  second  the  results  obtained  can  be  regarded  as  rough 
estimates  only.  The  value  of  h  in  these  formulas  is  really 
the  friction-head  hn  ',  since  in  their  deduction  the  other 
heads,  h'  ,  h"1  ',  and  /&"",  have  been  neglected  as  insensible. 
Hence  when  the  diameter  d,  the  length  /,  the  total  head  h, 
and  the  discharge  q  have  been  measured  for  a  long  pipe  the 
friction  factor  }  may  be  computed.  In  this  manner  much 
of  the  data  was  obtained  from  which  Table  33  has  been 
compiled. 

For  circular  orifices  and  for  short  tubes  of  equal  length 
under  the  same  head,  the  discharge  varies  as  the  square 
of  the  diameter.  For  pipes  of  equal  length  under  a  given 
head  the  discharges  vary  more  rapidly  owing  to  the  influence 
of  friction,  for  formula  (93)  2  shows  that  if  /  be  constant, 
q  varies  as  d$.  The  relative  discharging  capacities  of  pipes 
hence  vary  approximately  as  the  z\  powers  of  their  diam- 
eters. Thus,  if  two  pipes  of  diameters  dt  and  d2  have  same 
length  and  head,  and  if  ql  and  q2  be  their  discharges, 


For  example,  if  there  be  two  pipes  of  6  and  12  inches- 
diameter,  djd^  equals  2  and  hence  ^  =  5.7^,  or  the  second 
pipe  discharges  nearly  six  times  as  much  'as  the  first.  If 
the  variation  in  the  friction  factor  be  taken  into  account, 
the  formula  gives 


Now  as  the  values  of  /  vary  not  only  with  the  diameter 
but  with  the  velocity,  a  solution  cannot  be  made  except 
in  particular  cases.  For  the  above  example  let  the  ve- 
locity be  about  3  feet  per  second;  then  from  the  table 
fj  =0.023  and  /2  =0.019,  and  accordingly 

42  =^(2)^(1.  2)'  =6.  2^ 

or  the  1  2  -inch  pipe  discharges  more  than  six  times  as  much 
as  the  6-inch  pipe. 


ART.  94  PIEZOMETER    MEASUREMENTS  229 

Prob.  93a.  How  many  pipes,  6  inches  in  diameter,  are  equiv- 
alent in  discharging  capacity  to  one  pipe  24  inches  in  diameter? 

Prob  936.  Compute  the  diameter  required  to  deliver  15  ooo 
cubic  feet  per  hour  through  a  pipe  26  500  feet  long  under  a  head 
of  324.7  feet.  If  this  quantity  is  carried  in  two  pipes  of  equal 
diameter,  what  should  be  their  size? 


ART.  94.     PIEZOMETER  MEASUREMENTS 

Let  a  piezometer  tube  be  inserted  into  a  pipe  at  any  point 
D!  at  the  distance  /t  from  the  reservoir  measured  along  the 
pipe  line.  Let  AJ)±  be  the  vertical  depth  of  this  point 
below  the  water  level  of  Ai 

the  reservoir ;  then  if  the  "  " 
flow  be  stopped  at  the 
end  C,  the  water  rises  in 
the  tube  to  the  point  Ar 
But  when  the  flow  occurs, 
the  water  level  in  the  pi- 
ezometer stands  at  some 

point  C\,  and  the  pressure-head  at  Dl  is  hv  or  C1D1  in  the 
figure.  The  distance  Af^  then  represents  the  velocity- 
head  plus  all  the  losses  of  head  between  Dx  and  the  reser- 
voir. If  no  losses  of  head  occur  except  at  entrance  and  in 
friction,  the  value  of  A1Cl  then  is 


H  1=—  +  m—  +  M- 

2g  2g       'd    2g 

from  which  the  piezometric  height  can  be  found  when  v  has 
been  determined  by  direct  measurement  or  by  gaging. 

For  example,  let  the  total  length  £  =  3000  feet,  d  =  6 
inches,  h=q  feet,  and  7^=0.5.  Then,  as  in  Art.  89,  there 
is  found  /  =  0.026  and  ^  =  1.917  feet  per  second.  The 
position  of  the  top  of  the  piezometric  column  is  then  given  by 

Hl  =  (1.5  +  0.052^)  Xo. 05714 


230  FLOW  THROUGH  PIPES  CHAP,  vin 

and  the  height  of  that  column  above  the  pipe  is 


hl=AlDl-Hl 

Thus  if  /t  =  iooo  feet,  7^  =  3.06  feet;  and  if  ^  =  2000  feet, 
Hl  =6.03  feet.  If  the  pipe  is  so  laid  that  AlDl  is  9  feet,  the 
corresponding  pressure-heads  are  then  5.94  and  2.97  feet. 

For  a  second  piezometer  inserted  at  D2  at  the  distance  /2 
from  the  entrance,  the  value  of  H2  is 

v2         v2     L  v2 
#2=—  +  m—  +  /-§  — 

2        2g  2g       Jd2g 

Subtracting  from  this  the  expression  for  Hv  there  is  found 


The  second  member  of  this  formula  is  the  head  lost  in  fric- 
ti3n  in  the  length  /2  —  /t  (Art.  86),  and  the  first  member  is 
the  difference  of  the  piezometer  elevations.  Thus  is  again 
proved  the  principle  of  Art.  82,  that  the  difference  of  two 
piezometer  elevations  shows  the  head  lost  in  the  pipe  be- 
tween them;  in  Art.  82  the  elevations  Hl  and  H2  were 
measured  upward  from  the  datum  plane,  while  here  they 
have  been  measured  downward  from  the  water  level  in  the 
reservoir. 

By  the  help  of  this  principle  the  velocity  of  flow  in  a 
pipe  may  be  approximately  determined.  A  line  of  levels 
is  run  between  the  points  Dt  and  D2,  wr^ich  are  selected  so 
that  no  sharp  curves  occur  between  them,  and  thus  the 
difference  H2  —  Hl  is  found,  while  the  length  /2  —  /x  is  ascer- 
tained by  careful  chaining.  Then,  from  the  above  formula, 


-fr  W* 

i  ^2  ~~  y 

from  which  v  can  be  computed  by  the  help  of  the  friction 
factors  in  Table  33.     For  example,  Stearns,  in  1880,  made 


ART.  94  PIEZOMETER   MEASUREMENTS  231 

experiments  on  a  conduit  pipe  4  feet  in  diameter  under 
•different  velocities  of  flow.*  In  experiment  No.  2  the 
length  /2  —  /!  was  1747.2  feet,  and  the  difference  of  the 
piezometer  levels  was  1.243  feet.  Assuming  for  /  the  mean 
value  0.02,  and  using  32.16  feet  per  second  per  second  for 
g,  the  velocity  was 

(64.32  X  i.  243X4 

v  =  \-  -  =3.0  feet  per  second. 

X      0.02X1747 

This  velocity  in  the  table  of  friction  factors  gives  /  =  0.015 
for  a  4-foot  pipe.  Hence,  repeating  the  computation,  there 
is  found  v  =  3.50  feet  per  second  ;  it  is  accordingly  uncertain 
whether  the  value  of  /  is  0.015  or  0.014.  If  the  latter  value 
be  used,  there  is  found  v  =  3.62  feet  per  second.  The  actual 
velocity,  as  determined  by  measurement  of  the  water  over 
a  weir,  was  3.738  feet  per  second,  which  shows  that  the 
computation  is  in  error  about  4  percent. 

In  order  that  accurate  results  may  be  obtained  with 
piezometers  it  is  necessary,  particularly  under  low  pres- 
sure-heads, that  the  tubes  be  inserted  into  the  pipe  at 
right  angles.  If  they  be  inclined  with  or  against  the  cur- 
rent, the  pressure-head  h^  will  be  greater  or  less  than  that 
due  to  the  pressure  at  the  mouth.  Let  6  be  the  angle 
between  the  direction  of  the  flow  and  the  inserted  piezom- 
eter tube.  Since  the  impulse  in  the  direction  of  the  cur- 
rent is  proportional  to  the  velocity-head  (Art.  29),  the 
component  of  this  in  the  direction  of  the  inserted  tube 
tends  to  increase  the  normal  pressure-height  hl  when  6  is 
less  than  90°  and  to  decrease  it  when  6  is  greater  than  90°. 
Thus 


may  be  written  as  approximately  applicable  to  the  two 
cases  in  which  n  is  a  coefficient  whose  value  has  not  been 

*  Transactions  American  Society  of  Civil  Engineers,  1885,  vol.  14,  p.  4. 


232 


FLOW    THROUGH    PlPES 


CHAP.  VIII 


ascertained.  In  this,  if  the  tube  be  inserted  normal  to 
the  pipe,  6  =  90°  and  h0  becomes  hv  the  height  due  to  the 
static  pressure  in  the  pipe;  if  v  =  o,  the  angle  6  has  no 
effect  upon  the  piezometer  readings.  But  if  6  differs  from 


r     FIG.  946 

90°  by  a  small  angle,  the  error  in  the  reading  may  be  large 
when  the  velocity  in  the  pipe  is  high. 

The  question  as  to  the  point  from  which  the  pressure- 
head  should  be  measured  deserves  consideration.  In  the 
figures  of  preceding  articles  h^  and  h2  have  been  estimated 
upward  from  the  center  of  the  pipe,  and  it  is  now  to  be 
shown  that  this  is  probably  correct.  Let 
Fig.  94c  represent  a  cross-section  of  a  pipe 
to  which  are  attached  three  piezometers  as 
shown.  If  there  be  no  velocity  in  the  tube 
or  pipe,  the  water  surface  stands  at  the  same 
level  in  each  piezometer,  and  the  mean 
pressure-head  is  certainly  the  distance  of 
that  level  above  the  center  of  the  cross-section.  If  the 
water  in  the  pipe  be  in  motion,  probably  the  same  would 
hold  true.  Referring  to  formula  (73)!  and  to  Fig.  73a,  it  is 
also  seen  that  if  there  be  no  velocity  h' =h1  —  h2,  which 
cannot  be  true  unless  hl  —  h2=^oj  since  there  can  be  no  loss 
of  head  in  the  transmission  of  static  pressures;  hence  hv 
and  h2  cannot  be  measured  from  the  top  of  the  section.  In 
any  event,  since  the  piezometer  heights  represent  the  mean 
pressures,  it  appears  that  they  should  be  reckoned  upward 
from  the  center  of  the  section.  The  piezometer  couplings 
for  hose  devised  by  Freeman  are  arranged  with  connections 


FIG.  94c 


ART.  95  THE    HYDRAULIC    GRADIENT  233 

on  the  top,  bottom,  and  sides,  as  are  also  those  used  for 
the  Venturi  meter  (Art.  38),  and  thus  the  results  obtained 
correspond  to  mean  pressures  or  pressure-heads.  Even  in 
cases  where  the  two  points  of  connection  are  so  near  together 
that  the  difference  H2  —  Hl  can  be  measured  by  a  differential 
manometer  (Art.  37),  the  method  of  connecting  the  tubes 
to  the  pipes  should  receive  careful  attention. 

Prob.  94a.  To  a  pipe  of  uniform  size  two  piezometers  are 
attached  at  points  A  and  B  one  mile  apart,  the  point  A  being 
82.13  feet  higher  than  B.  The  pressure-heads  read  simul- 
taneously at  the  two  stations  are  6.07  feet  for  A  and  88.21  feet 
for  B.  In  which  direction  does  the  water  flow? 

Prob.  946.  At  a  point  500  feet  from  the  reservoir,  and  28 
feet  below  its  surface,  a  pressure  gage  reads  10.5  pounds  per 
square  inch;  at  a  point  8500  feet  from  the  reservoir  and  280.5 
feet  below  its  surface,  it  reads  61  pounds  per  square  inch.  If 
the  pipe  be  12  inches  in  diameter,  compute  the  discharge. 


ART.  95.     THE  HYDRAULIC  GRADIENT 

The  hydraulic  gradient  is  a  line  which  connects  the 
water  levels  in  piezometers  placed  at  intervals  along  the 
pipe ;  or  rather,  it  is  the  line  to  which  the  water  levels  would 
rise  if  piezometer  tubes  A 

were  inserted.  In  Fig.  94a  '  "~f~ 
the  line  EC  is  the  hydraulic 
gradient,  and  it  is  now  to 
be  shown  that  for  a  pipe 
of  uniform  size  this  is  ap- 
proximately a  straight  line. 
For  a  pipe  discharging 

freely  into  the  air,  as  in  Fig.  94a,  this  line  joins  the  outlet 
end  with  a  point  B  near  the  top  of  the  reservoir.  For  a 
pipe  with  submerged  discharge,  as  in  Fig.  95a,  it  joins  the 
lower*  water  level  witli  the  point  B. 

Let  Dl  be  any  point  on  the  pipe  distant  /t  from  the  reser- 


234  FLOW   THROUGH   PIPES  CHAP.  VIII 

voir,  measured  along  the  pipe  line.  The  piezometer  there 
placed  rises  to  Cv  which  is  a  point  in  the  hydraulic  gradient. 
The  equation  of  this  line  with  reference  to  the  origin  A  is 
given  by  the  first  equation  of  Art.  94, 


H 


in  which  Hl  is  the  ordinate  Af^  and  lt  is  the  abscissa 
AAlt  provided  that  the  length  of  the  pipe  is  sensibly  equiva- 
lent to  its  horizontal  projection.  In  this  equation  the 
first  term  of  the  second  member  is  constant  for  a  given 
velocity,  and  is  represented  in  the  figure  by  A  B  or  A^B^\ 
the  second  term  varies  with  /lf  and  is  represented  by 
Bfr  The  gradient  is  therefore  a  straight  line,  subject 
to  the  provision  that  the  pipe  is  laid  approximately  hori- 
zontal; which  is  usually  the  case  in  practice,  since  quite 
material  vertical  variations  may  exist  in  long  pipes  with- 
out sensibly  affecting  the  horizontal  distances. 

When  the  variable  point  D1  is  taken  at  the  outlet  end 
of  the  pipe,  Hl  becomes  the  head  h,  and  /x  becomes  the 
total  length  /,  agreeing  with  the  formula  of  Art.  89,  if 
the  losses  of  head  due  to  curvature  and  valves  be  omitted. 
When  dt  is  taken  very  near  the  inlet  end,  lt  becomes  zero 
and  the  ordinate  Hl  becomes  AB,  which  represents  the 
velocity-head  plus  the  loss  of  head  at  entrance. 

When  there  are  easy  horizontal  curves  in  a  pipe  line, 
the  above  conclusions  are  unaffected,  except  that  the 
gradient  BC  is  always  vertically  above  the  pipe,  and 
therefore  can  be  called  straight  only  by  courtesy,  although 
as  before  the  ordinate  BlCl  is  proportional  to  lr  When 
there  are  sharp  curves,  the  inclination  of  hydraulic  gradient 
becomes  greater  and  it  is  depressed  at  each  curve  by  an 
amount  equal  to  the  loss  of  head  which  there  occurs. 
When  an  obstruction  occurs  in  a  pipe  or  a  valve  is  'par- 
tially closed  there  is  a  sudden  depression  of  the  gradient. 


ART.  95 


THE  HYDRAULIC  GRADIENT 


235 


FIG.  956 


If  the  pipe  is  so  laid  that  a  portion  of  it  rises  above 
the   hydraulic   gradient   as   at   D1   in   Fig.    956,    an   entire 
change    of    condition    generally    results.     If    the    pipe    be 
closed   at  C,  all  the  piez- 
ometers stand  in  the  line 
A  A,    at    the    same    level 
as     the     surface     of    the 
reservoir.        When      the 
valve  at  C  is  opened,  the 
flow  at  first  occurs  under 
normal  conditions,  h  being 
the  head  and  BC  the  hydraulic  gradient.     The  pressure- 
head  at  DI  is  then  negative,   and  represented  by  DlCr 
As  a  consequence  air  tends  to  enter  the  pipe,  and  when 
it  does  so,  owing  to  defective  joints,  the  continuity  of  the 
flow  is  broken,  and  then  the  pipe  from  Dl  to  C  is  only 
partly  filled  with  water.     The  hydraulic  gradient  is  then 
shifted  to  BDV  the  discharge  occurs  at  Dl  under  the  head 
AJ)V  while  the  remainder  of  the  pipe  acts  merely  as  a 
channel   to    deliver   the    flow.     It    usually    happens    that 
this  change  results  in  a  great  diminution  of  the  discharge, 
so  that  it  has  been  necessary  to  dig  up  and  relay  portions 
of  a  pipe  line  which  have  been  inadvertently  run  above 
the    hydraulic    gradient.     This    trouble    can    always    be 
avoided   by  preparing    a   profile    of   the    proposed   route, 
drawing  the  hydraulic  gradient  upon  it,   and  excavating 
the  pipe  trench  well  below  the  gradient.     In  cases  where 
the  cost  of  this  excavation  is  so  great  that  it  is  resolved 
to  lay  the  pipe  above  the  gradient,  all  the  joints  of  the 
pipe  above  the  gradient  should  be  made  absolutely  tight 
so  that  no  air  can  enter. 

When  a  large  part  of  the  pipe  lies  above  the  hydraulic 
gradient  it  is  called  a  siphon.  Conditions  sometimes  exist 
which  require  a  pipe  line  to  be  laid  as  a  siphon  for  a  short 
distance.  In  such  a  case  an  air  chamber  is  sometimes 
built  at  the  highest  elevation  so  that  air  may  collect 


236  FLOW  THROUGH  PIPES  CHAP,  vin 

in  it  instead  of  in  the  pipe,  and  provision  is  made  for 
recharging  the  siphon  when  the  flow  ceases  by  admitting 
water  at  the  highest  elevation,  or  by  operating  a  suction- 
pump  placed  there,  or  by  forcing  water  into  the  pipe 
by  a  pump  located  at  a  lower  elevation.  Probably  the 
largest  siphon  ever  constructed  is  that  laid  about  1885 
at  Kansas  City,  Mo.,  it  being  42  inches  in  diameter,  and 
730  feet  long,  with  the  summit  10  feet  above  the  general 
level  of  the  pipe  line.  The  air  that  collected  at  the  summit 
was  removed  by  operating  a  steam  ejector  for  a  few 
minutes  each  day.* 

The  pressure-head  hl  at  any  distance  /t  from  the  reservoir 
may  be  expressed  in  terms  of  the  total  head  h  by  an  in- 
spection of  Fig.  95a  where  Af)^  is  the  hydrostatic  head 
Hl  and  ClDl  is  hr  Thus  h1=Hl-AlCl  and,  since  AlBl 
is  very  small  for  long  pipes,  the  similar  triangles  give 


This  result  can  also  be  obtained  from  the  above  formula  for 
H^  by  making  i+m  equal  to  zero  and  placing  for  v  its 
value  from  (86).  The  loss  of  head  in  friction  is  represented 
in  the  figure  by  Bfv  and  the  value  of  this  is  (ljl)h  for 
long  pipes  ;  that  is,  '  this  loss  of  head  is  proportional  to 
the  length  of  the  pipe. 

The  above  discussion  shows  that  it  is  immaterial  where 
the  pipe  enters  the  reservoir,  provided  that  it  enters 
below  the  hydraulic  gradient  point  B.  It  is  also  not  to 
be  forgotten  that  the  whole  investigation  rests  on  the 
assumption  that  the  lengths  /x  and  /  are  sensibly  equal 
to  their  horizontal  projections. 

Prob.  95.  A  pipe  3  inches  in  diameter  discharges  538  cubic 
feet  per  hour  under  a  head  of  12  feet.  At  a  distance  of  300 
feet  from  the  reservoir  the  depth  of  the  pipe  below  the  water 
surface  in  the  reservoir  is  4.5  feet.  Compute  the  probable 
pressure-head  at  this  point. 

*  Engineering  News,  1891,  vol.  26,  p.  519;   1893,  vol.  29,  pp.  423,  588. 


ART.  96  A   COMPOUND    PlPE  237 


ART.  96.     A  COMPOUND  PIPE 

A  compound  pipe  is  one  having  different  sizes  in 
different  portions  of  its  length.  The  change  from  one 
length  to  another  should  be  made  by  a  '  *  reducer, ' '  which 
is  a  conical  frustum 
several  feet  long,  so  that 
losses  of  head  due  to  sud- 
den enlargement  or  con- 
traction are  avoided  (Art. 
74).  Let  dv  dv  dv  etc.,  FIG.  96 

be  the  diameters;  llt  lv  19,  etc.,  the  corresponding  lengths, 
the  total  length  being  /1  +  /2  +  etc.  Let  vv  vv  etc.,  be 
the  velocities  in  the  different  sections.  Neglecting  the 
loss  of  head  at  entrance  and  also  that  lost  in  curvature, 
the  total  head  h  may  be  placed  equal  to  the  loss  of  head 
in  friction,  or 


•  ia1  2g    '  *a2  2g 
Now  if  the  discharge  per  second  be  q,  and  the  flow  be  steady 

Substituting  these  velocities  and  solving  for  q,  gives 

(96) 


in  which  the  friction  factors  flt  fv  etc.,  corresponding  to 
the  given  diameters  and  the  computed  velocities  are  to  be 
taken  from  Table  33. 

For  example,  consider  the  case  of  a  pipe  having  only 
two  sizes ;  let  dt  =  2  and  /t  =  2800  feet,  d2  =  1.5  and  12  =  2 145 
feet,  and  ^=127.5  feet.  Using  for  fl  and  /2  the  mean 


238  FLOW  THROUGH  PIPES  CHAP,  vin 

value  0.02,  and  making  the  substitutions  in  the  formula, 
there  is  found 

2  =  26.2  cubic  feet  per  second, 
from  which     ^  =  8.3    and    v2  =  14.  8  feet  per  second. 

Now  from  Table  33  it  is  seen  that  ^=0.015  and  /2  =  0.015; 
and  repeating  the  computation, 

'2  =  30.2  cubic  feet  per  second, 
which  gives    vl  =  9.6     and    v2  =  17.1  feet  per  second. 

These  results  are  probably  as  definite  as  the  table  of  friction 
factors  will  allow,  but  are  to  be  regarded  as  liable  to  an  un- 
certainty of  several  percent. 

To  determine  the  diameter  of  a  pipe  which  will  give  the 
same  discharge  as  the  compound  one,  it  is  only  necessary  to 
replace  the  denominator  in  the  above  value  of  q  by  fl/d5, 
where  I  =/1  +  /2  +  etc.,  and  d  is  the  diameter  required.  Tak- 
ing the  values  of  /  as  equal,  this  gives 

* 


Applying  this  to  the  above  example,  it  becomes 

4945^2800     2145 
d5          25     h  i.s5 

from  which  d  =  1.68  feet,  or  about  20  inches. 

A  compound  pipe  is  sometimes  used  to  prevent  the 
hydraulic  gradient  from  falling  below  the  pipe  line.  Thus, 
it  is  seen  in  Fig.  96  that  the  hydraulic  gradient  rises  at  Dt 
and  falls  at  D2,  and  that  its  slope  over  the  larger  pipe  is 
less  than  over  the  smaller  one.  These  slopes  and  amount 
of  rise  at  Dl  can  be  computed  for  a  given  case.  Using  the 
above  numerical  data  the  loss  of  head  in  friction  for  100 
feet  of  the  large  pipe  is 

100  V 
&"—  p.o'is  --  —  =  1.07  feet 

2       2g  ' 


ART.  96  A  COMPOUND    PlPE  239 

while  the  same  for  the  small  pipe  is  4.55  feet.  Hence  the 
slope  of  the  gradients  ACl  and  Cf  is  more  than  four  times 
as  rapid  as  that  of  the  gradient  EJLy  In  the  large  pipe 
at  D1  the  velocity-head  is  0.01555  XQ.62  =  1.43  feet,  and, 
supposing  that  no  loss  occurs  in  the  reducer,  the  velocity- 
head  for  the  small  pipe  is  4.55  feet.  The  vertical  rise  ClEl 
of  the  hydraulic  gradient  at  Dl  is  hence  the  rise  in  pressure- 
head  4.55  —  1.43=3.12  feet,  and  a  fall  of  equal  amount 
occurs  at  D2. 

When  a  portion  of  a  small  pipe  is  to  be  replaced  by  a 
large  one  it  is  immaterial  in  what  part  of  the  length  it  is 
introduced,  for  it  is  seen  that  formula  (96)  takes  no  note 
of  where  the  length  /t  is  placed  in  the  total  distance  /.  The 
Romans  knew  that  an  increase  in  the  diameter  of  a  pipe 
after  leaving  the  reservoir  would  increase  the  discharge, 
and  the  law  passed  by  the  Roman  senate  about  the  year 
10  B.C.  forbade  a  consumer  to  attach  a  larger  pipe  to  the 
standard  pipe  within  50  feet  of  the  reservoir  to  which  the 
latter  was  connected.* 

Prob.  96a.  A  pipe  400  feet  long  leads  from  a  reservoir  to  a 
house,  the  first  50  feet  being  i  inch  and  the  remainder  2  inches 
in  diameter.  Compute  the  discharge  in  gallons  per  minute  for 
a  head  of  16  feet.  Compute  also  the  discharge  if  the  entire 
length  be  i  inch  in  diameter.  Draw  the  hydraulic  gradient  for 
each  case. 

Prob.  966.  At  Rochester,  N.  Y.,  there  is  a  pipe  102  277  feet 
long,  of  which  50  828  feet  is  36  inches  in  diameter  and  51  449 
feet  is  24  inches  in  diameter.  Under  a  head  of  143.8  feet  this 
pipe  is  said  to  have  discharged  in  1876  about  14  cubic  feet  per 
second  and  in  1890  about  10^  cubic  feet  per  second.  Compute 
the  discharge  by  (96),  and  draw  the  hydraulic  gradient. 

Prob.  96c.  Frontinus  said  that  the  diameter  and  circum- 
ference of  a  denaria  pipe  (Art.  85)  in  digits  were  2+£  and 
T+i  +  iV  +  'sIff  an(i  that  it  had  a  capacity  of  4  quinarias.  What 
value  of  TT  did  Frontinus  probably  use  in  his  computations  ? 

*  Herschel,  Water  Supply  of  the  City  of  Rome  (Boston,  1889),  p.  77. 


240  FLOW   THROUGH    PIPES  CHAP.  VIII 

ART.  97.     A  PIPE  WITH  A  NOZZLE 

Water  is  often  delivered  through  a  nozzle  in  order  to 
perform  work  upon  a  motor  or  for  the  purposes  of  hydraulic 
mining,  the  nozzle  being  attached  to  the  end  of  a  pipe 
which  brings  the  flow  from  a  reservoir.  In  such  a  case  it 

is  desirable  that  the  pressure 
at  the  entrance  to  the  nozzle 
=^  should  be  as  great  as  possi- 
ble, and  this  will  be  effected 
when  the  loss  of  head  in  the 
pipe  is  as  small  as  possible. 
'• 97  The    pressure    column    in    a 

piezometer,  supposed  to  be  inserted  at  the  end  of  the  pipe, 
as  shown  at  CJ)l  in  Fig.  97,  measures  the  pressure-head 
there  acting,  and  the  height  Af^  measures  the  lost  head 
plus  the  velocity-head,  the  latter  being  very  small. 

Let  h  be  the  total  head  on  the  end  of  the  nozzle,  D  its 
diameter,  and  V  the  velocity  of  the  issuing  stream.  Let 
d  and  v  be  the  corresponding  quantities  for  the  pipe,  and  / 
its  length.  Then  the  effective  velocity-head  of  the  issuing 
stream  is  V2/2g  and  the  lost  head  is  h—  V2/2g.  This  lost 
head  consists  of  several  parts — that  lost  at  the  entrance 
D',  that  lost  in  friction  in  the  pipe;  that  lost  in  curves 
and  valves,  if  any ;  and  lastly,  that  lost  in  the  nozzle.  Thus 

h-  —  -    v^     f-v~         ^1         ^        '— 

2g  2g         ~d2g  I2g  22g  2g 

Here  m  is  determined  by  Art.  85,  /  by  Art.  86,  ml  by  Art. 
87,  w2  by  Art.  88,  while  m'  for  the  nozzle  is  found  in  the 
same  manner  as  m  is  found  for  the  pipe,  or  m'  =  (i/^)2—  i, 
where  £,  is  the  coefficient  of  velocity  for  the  nozzle  (Art.  80). 
This  value  of  m'  takes  account  of  all  losses  in  the  nozzle,  so 
that  it  is  unnecessary  to  consider  its  length;  for  a  perfect 
nozzle  Cj  is  unity  and  m'  is  zero. 


ART.  97  A    .flPE    WITH    A    NOZZLE  241 

The  velocities  v  and  V  are  inversely  as  the  areas  of  the 
corresponding  cross-sections  (Art.  32),  since  the  flow  is 
steady,  whence  V=v(d/D)2.  Inserting  this  in  the  above 
equation  and  solving  for  v  gives,  if  ml  and  m2  be  neglected, 


(97) 


for  the  velocity  in  the  pipe.  The  velocity  and  discharge 
from  the  nozzle  are  then  given  by 

V  =  (d/D)2v  q  =  \nD2V 

and  the  velocity-head  of  the  jet  is  V2/2g.  These  equations 
show  that  the  greatest  value  of  V  obtains  when  D  is  as 
small  as  possible  compared  to  d,  and  that  the  greatest  dis- 
charge occurs  when  D  is  equal  to  d.  When  the  object  of 
a  nozzle  is  to  utilize  the  velocity-head  of  a  jet,  a  large  pipe 
and  a  small  nozzle  should  be  employed.  When  the  object 
is  to  utilize  the  energy  of  the  jet  in  producing  power  by  a 
water  wheel,  there  is  a  certain  relation  between  D  and  d 
that  renders  this  a  maximum  (Art.  161). 

As  a  numerical  example,  the  effect  of  attaching  a  nozzle 
to  the  pipe  whose  discharge  was  computed  in  Art.  90  will 
be  considered.  There  /  =  i5oo,  ^  =  0.25,  and  ^=64  feet; 
w  =  o.5,  ^  =  5-3  feet,  and  3  =  0.26  cubic  feet  per  second. 
Now  let  the  nozzle  be  one  inch  in  diameter  at  the  small 
end,  or  D  =0.0833  feet,  and  let  its  coefficient  ct  be  0.98. 
Here  d/D  =  3,  and  for  /  =  0.025  ^ne  velocity  in  the  pipe  is 


2X32.16X64 


0.5+0.025  X 1 500X4  +  1. 041  X8i 

or  v  =  4. 2  feet  per  second.  The  effect  of  the  nozzle,  therefore, 
is  to  reduce  the  velocity  in  the  pipe.  The  velocity  of  the 
jet  at  the  end  of  the  nozzle  is,  however, 

V  =v(d/D)2  =  37.8  feet  per  second 


242  FLOW  THROUGH  PIPES  CHAP,  vin 

and  the  discharge  per  second  from  the  nozzle  is 
q  =  ^nDzV  ^=0.206  cubic  feet 

which  is  about  30  percent  less  than  that  of  the  pipe  before 
the  nozzle  was  attached.  The  nozzle,  however,  produces 
a  marvellous  effect  in  increasing  the  energy  of  the  discharge ; 
for  the  velocity-head  corresponding  to  5.3  feet  per  second 
is  only  0.44  feet,  while  that  corresponding  to  37.8  feet  per 
second  is  22.2  feet,  or  about  50  times  as  great.  As  the 
total  head  is  64  feet,  the  efficiency  of  the  pipe  and  nozzle 
is  about  35  percent. 

If  the  pressure-head  hl  at  the  entrance  of  the  nozzle  be 
observed,  either  by  a  piezometer  tube  or  by  a  pressure  gage, 
the  velocity  of  discharge  from  the  nozzle  can  be  computed 
by  the  formula 


the  demonstration  of  which  is  given  in  Art.  80.  This  can 
be  used  when  a  hose  and  nozzle  is  attached  at  any  point  of 
a  pipe  or  at  a  hydrant.  It  can  also  be  used  to  compute  ht 
when  V  has  been  found.  Thus,  for  the  above  example, 

/i      D*\V2 
k=(-^ — n- 1— -22.8  feet 


which  shows  that  the  loss  of  head  in  the  nozzle  is  about  0.6 
feet.  The  loss  of  head  at  entrance,  for  this  case,  is  0.2  feet, 
and  the  loss  of  head  in  friction  in  the  pipe  is  41.0  feet. 

Prob.  97a.  Compute  the  velocity,  discharge,  velocity-head, 
and  friction-head  for  a  pipe  and  nozzle,  taking  the  data  of  the 
above  numerical  example,  except  that  1=2 500  feet. 

Prob.  976.  A  pipe  12  inches  in  diameter  and  4320  feet  long 
leads  from  a  reservoir  to  a  gravel  bank  against  which  water  is 
delivered  from  a  nozzle  2  inches  in  diameter.  The  head  on  the 
end  of  the  nozzle  is  320  feet  and  the  coefficient  of  velocity  of  the 
nozzle  is  0.97.  Compute  the  velocity  in  the  pipe,  the  velocity- 
head  of  the  jet,  and  the  discharge. 


ART.  98  HOUSE-SERVICE    PlPES  243 

ART.  98.     HOUSE-SERVICE  PIPES 

A  service  pipe  which  runs  from  a  street  main  to  a  house 
is  connected  to  the  former  at  right  angles,  and  usually 
by  a  '  *  ferrule ' '  which  is  smaller  in  diameter  than  the  pipe 
itself.  The  loss  of  head 
at  entrance  is  hence  larger 
than  in  the  cases  before 
discussed,  and  m  should 
probably  be  taken  as  at 
least  equal  to  unity.  The 
pipe,  if  of  lead,  is  frequently 

carried   around   sharp   cor- 

-  £11 

ners    by    curves    of    small 

radius;  if  of  iron,  these  curves  are  formed  by  pieces  form- 
ing a  quadrant  of  a  circle  into  which  the  straight  parts 
are  screwed,  the  radius  of  the  center  line  of  the  curve 
being  but  little  larger  than  the  radius  of  the  pipe,  so  that 
each  curve  causes  a  loss  of  head  equal  nearly  to  double 
the  velocity-head  (Art.  87).  For  new  iron  pipes  the  loss 
of  head  due  to  friction  may  be  estimated  by  the  rules  of 
Art.  86  or  by  Table  35. 

A  water  main  should  be  so  designed  that  a  certain 
minimum  pressure-head  h^  exists  in  it  at  times  of  heaviest 
draft.  This  pressure-head  may  be  represented  by  the 
height  of  the  piezometer  column  AB,  which  would  rise 
in  a  tube  supposed  to  be  inserted  in  the  main,  as  in  Fig. 
98a.  The  head  h  which  causes  the  flow  in  the  pipe  is  then 
the  difference  in  level  between  the  top  of  this  column 
and  the  end  of  the  pipe,  or  AC.  Inserting  for  h  this  value, 
the  formulas  of  Arts.  90  and  91  may  be  applied  to  the 
investigation  of  service  pipes  in  the  manner  there  illus- 
trated. As  the  sizes  of  common  house-service  pipes  are 
regulated  by  the  practice  of  the  plumbers  and  by  the 
market  sizes  obtainable,  it  is  not  often  necessary  to  make 
computations  regarding  them. 


244  FLOW  THROUGH  PIPES  CHAP,  vin 

The  velocity  of  flow  in  the  main  has  no  direct  influence 
upon  that  in  the  pipe,  since  the  connection  is  made  at 
right  angles.  But  as  that  velocity  varies,  owing  to  the 
varying  draft  upon  the  main,  the  effective  head  h  is 
subject  to  continual  fluctuations.  When  there  is  no 
flow  in  the  main,  the  piezometer  column  rises  until  its 
top  is  on  the  same  level  as  the  surface  of  the  reservoir; 
in  times  of  great  draft  it  may  sink  below  C,  so  that 
no  water  can  be  drawn  from  the  service  pipe. 

The  detection  and  prevention  of  the  waste  of  water 
by  consumers  is  a  matter  of  importance  in  cities  where 
the  supply  is  limited  and  where  meters  are  not  in  use. 
Of  the  many  methods  devised  to  detect  this  waste,  one 
by  the  use  of  piezometers  may  be  noticed,  by  which  an 
inspector  without  entering  a  house  may  ascertain  whether 
water  is  being  drawn  within,  and  the  approximate  amount 
per  second.  Let  M  be  the  street  main  from  which  a 
service  pipe  MOH  runs  to  a  house  H.  At  the  edge  of  the 
sidewalk  a  tube  OP  is  connected  to  the  service  pipe,  which 
has  a  three-way  cock  at  0,  which  can 
be  turned  from  above.  The  inspector, 
passing  on  his  rounds  in  the  night-time, 
attaches  a  pressure  gage  at  P  and 
turns  the  cock  0  so  as  to  shut  off  the 
water  from  the  house  and  allow  the 
full  pressure  of  the  main  pl  to  be  registered.  Then  he 
turns  the  cock  so  that  the  water  may  flow  into  the  house, 
while  it  also  rises  in  OP  and  registers  the  pressure  py 
Then  if  p2  is  less  than  pl  it  is  certain  that  waste  is  occurring 
within  the  house,  and  the  amount  of  this  may  be  approxi- 
mately computed,  if  desired,  and  the  consumer  be  fined 
accordingly. 

The  pitometer,  which  consists  of  a  rated  Pitot  tube 
(Art.  41)  facing  the  current  in  the  pipe,  with  a  differential 
gage  (Art.  37)  to  determine  the  pressure-head  clue  to 
the  current,  is  also  used  for  the  measurement  cf  the  flow 


ART.  98  HOUSE-SERVICE  PIPES  245 

in  water  mains  and  for  the  detection  of  water  waste. 
A  photographic  record  of  the  difference  in  height  of  the 
columns  of  liquid  in  the  gage  tube  is  kept,  and  this  shows 
the  discharge  through  the  water  main  at  any  instant, 
as  also  all  fluctuations  in  the  flow.* 

When  the  pressure  in  the  street  main  is  very  high, 
a  pressure  regulator  may  be  placed  between  the  main 
and  the  house  in  order  to  reduce  the  pressure  and  thus 
allow  lighter  pipes  to  be  used  in  the  house.  Fig.  98c 
shows  the  principle  of  its  action,  where  A  represents 
the  pipe  from  the  main  and  B  the  pipe  leading  to 
the  house.  A  weight  W  is  placed  upon  a  piston  which 
covers  the  opening  into  the  cham- 
ber C.  This  weight  and  that  of 
the  piston  is  sufficient  to  over- 
come a  certain  unit-pressure  in  C 
and  therefore  the  unit-pressure  in 

B  is  less  than  that   in  A  by  that 

^  -  FIG.  98c 

amount.       For   example,    suppose 

the  pressure  in  A  to  be  100  pounds  per  square  inch,  and  let  it 
be  required  that  the  pressure  in  B  shall  not  rise  above  60 
pounds  per  square  inch ;  then  the  piston  must  be  so  weighted 
that  it  may  exert  on  the  water  in  C  a  pressure  of  40  pounds 
per  square  inch.  If  water  be  drawn  out  anywhere  along 
the  pipe  B  the  pressure  in  the  chamber  above  the  piston 
falls  below  60  pounds  per  square  inch,  and  hence  the 
piston  rises  and  water  flows  from  A  into  B  until  the  pres- 
sure is  restored.  Instead  of  a  weight,  a  spring  is  generally 
used,  or  sometimes  a  weighted  lever. 

Prob.  98.  In  Fig.  986  let  the  house  pipe  be  one  inch  in  diam- 
eter and  the  pressure  at  the  gage  be  32  pounds  per  square  inch 
when  there  is  no  flow.  The  distance  from  the  main  to  the  gage 
is  1 5  feet  and  from  the  gage  to  the  end  of  the  pipe  is  28  feet.  At. 
the  end  of  the  pipe,  which  is  4  feet  higher  than  the  gage,  1.8 
gallons  of  water  are  drawn  per  minute.  Compute  the  pressure; 
at  the  gage. 

*  Engineering  Record,  1903,  vol.  47,  p.  122. 


246  FLOW  THROUGH  PIPES  CHAP,  vin 


ART.  99.     WATER  MAINS  IN  TOWNS 

The  simplest  case  of  the  distribution  of  water  is  that 
where  a  single  main  is  tapped  by  a  number  of  service 
pipes  near  its  end,  as  shown  in  Fig.  99.  In  designing  such 

a  main  the  principal  consider- 
ation is  that  it  should  be  large 
enough  so  that  the  pressure- 
head  hv  when  all  the  pipes 
are  in  draft,  shall  be  amply 
sufficient  to  deliver  the  water 
FIG  Q9  into  the  highest  houses  along 

the  line.  It  is  generally  recom- 
mended that  this  pressure-head  in  commercial  and  manu- 
facturing districts  should  not  be  less  than  150  feet,  and 
in  suburban  districts  not  less  than  100  feet.  The  height 
H  to  the  surface  of  the  water  in  the  reservoir  will  always 
be  greater  than  hiy  and  the  pipe  is  to  be  so  designed  that 
the  losses  of  head  may  not  reduce  h^  below  the  limit 
assigned.  The  head  h  to  be  used  in  the  formulas  is  the 
difference  H  —  hr  The  discharge  per  second  q  being 
known  or  assumed,  the  problem  is  to  determine  the  diam- 
eter d  of  the  main. 

A  strict  theoretical  solution  of  even  this  simple  case 
leads  to -very  complicated  calculations,  and  in  fact  cannot 
be  made  without  knowing  all  the  circumstances  regarding 
each  of  the  service  pipes.  Considering  that  the  result 
of  the  computation  is  merely  to  enable  one  of  the  market 
sizes  to  be  selected,  it  is  plain  that  great  precision  cannot 
be  expected,  and  that  approximate  methods  may  be  used 
to  give  a  solution  entirely  satisfactory.  It  will  then  be 
assumed  that  the  service  pipes  are  connected  with  the  main 
at  equal  intervals,  and  that  the  discharge  through  each 
is  the  same  under  maximum  draft.  The  velocity  v  in 
the  main  then  decreases  and  becomes  o  at  the  dead 


ART.  99  WATER  MAINS  IN  TOWNS  247 

end.  The  loss  of  head  per  linear  foot  in  the  length  /t 
(Fig.  99)  is  hence  less  than  in  /.  To  determine  the  total 
loss  of  head  in  the  length  ll}  let  vl  be  the  velocity  at  a  dis- 
tance x  from  the  dead  end  ;  then  vl—v.  x/lt  and  the  loss  of 
head  in  friction  in  the  length  dx  is 


'  d  2g     '  dl^  2g 
and  hence  between  the  limits  o  and  ^  that  loss  of  head  .is 


provided  that  /  remains  constant.  This  is  really  not 
the  case,  but  no  material  error  is  thus  introduced,  since 
/  must  be  taken  larger  than  the  tabular  values  in  order 
to  allow  for  the  deterioration  of  the  inner  surface  of  the 
main.  The  loss  of  head  in  friction  for  a  pipe  which  dis- 
charges uniformly  along  its  length  may  therefore  be  taken 
at  one-third  of  that  which  occurs  when  the  discharge 
is  entirely  at  the  end. 

Now  neglecting  the  loss  of  head  at  entrance  and  the 
effective  velocity-head  of  the  discharge,  the  total  head  h 
is  entirely  consumed  in  friction,  or 

h=fl—     il±  — 
~'d2g     '  $d  2g 

Placing  in  this  for  v  its  value  in  terms  of  the  total  discharge 
q  and  the  diameter  of  the  pipe,  and  solving  for  d,  gives 


This  is  the  same  as  the  formula  of  Art.  93,  except  that  /  has 
been  replaced  by  /  +  J/r     The  diameter  in  feet  then  is 


when  h  and  /  are  in  feet  and  q  in  cubic  feet  per  second. 


248  FLOW  THROUGH  PIPES  CHAP,  vin 

For  example,  consider  a  village  consisting  of  a  single 
street  with  length  Zt  =  3000  feet,  and  upon  which  there  are 
TOO  houses,  each  furnished  with  a  service  pipe.  The  prob- 
able population  is  then  500,  and  taking  100  gallons  per  day 
as  the  consumption  per  capita,  this  gives  for  the  average 
discharge  per  second  along  the  length  /x 

500X100 

q  = £—  —  =0.0774  cubic  feet, 

7.48X3600X24 

and  since  the  maximum  draft  is  often  double  of  the 
average,  q  will  be  taken  as  0.15  cubic  feet  per  second.  The 
length  /  to  the  reservoir  is  4290  feet,  whose  surface  is  90.5 
feet  above  the  dead  end  of  the  main,  and  it  is  required  that 
under  full  draft  the  pressure-head  in  the  main  shall  be 
75  feet.  Then  ^=90.5  —  75  =  15.5  feet,  and  taking  /  =  o.o3 
in  order  to  be  on  the  safe  side,  the  formula  gives 

^  =  0.36  feet  =4. 3  inches. 

Accordingly  a  four-inch  pipe  is  nearly  large  enough  to  sat- 
isfy the  imposed  conditions. 

To  consider  the  effect  of  fire  service  upon  the  diameter 
of  the  main,  let  there  be  four  hydrants  placed  at  equal  in- 
tervals along  the  line  llt  each  of  which  is  required  to  deliver 
20  cubic  feet  per  minute  under  the  same  pressure-head  of 
75  feet.  This  gives  a  discharge  1.33  cubic  feet  per  second, 
or,  in  total,  9  =  1.33  +  0.15  =  1.5  cubic  feet.  Inserting  this 
in  the  formula,  and  using  for  /  the  same  value  as  before, 

^  =  0.897  feet  =  10.8  inches. 

Hence  a  ten-inch  pipe  is  at  least  required  to  maintain  the 
required  pressure  when  the  four  hydrants  are  in  full  draft 
at  the  same  time  with  the  service  pipes. 

Prob.  99.  Compute  the  velocity  v  and  the  pressure-head  h^ 
for  the  above  example,  if  the  main  be  10  inches  in  diameter  and 
the  discharge  be  1.5  cubic  feet  per  second.  Also  when  the 
main  is  12  inches  in  diameter. 


ART.  100 


BRANCHES  AND  DIVERSIONS 


249 


ART.  100.     BRANCHES  AND  DIVERSIONS 

In  Fig.  lOOa  is  shown  a  main  of  length  /  and  diameter  d, 
connected  with  a  storage  reservoir,  which  has  two  branches 
with  lengths  /x  and  /2,  and  diameters  dt  and  d2  leading  to 


FIG.  lOOa 

two  smaller  distributing  reservoirs.  These  data  being  given, 
as  also  the  heads  H^  and  H2  under  which  the  flow  occurs, 
it  is  required  to  find  the  discharges  ql  and  q2.  Let  v,  vl9 
and  v2  be  the  corresponding  velocities  ;  then  for  long  pipes, 
in  which  all  losses  except  those  due  to  friction  may  be 
neglected,  the  friction-heads  for  the  two  branches  are 


where  y  is  the  difference  in  level  between  the  reservoir  sur- 
face and  the  surface  of  the  water  in  a  piezometer  tube 
supposed  to  be  inserted  at  the  junction.  This  y  is  the 
friction -head  consumed  in  the  flow  in  the  large  main,  and 
hence  from  formula  (86)  its  value  is 

A    v2 


Inserting  this  in  the  two  equations,  and  placing  for  the  ve- 
locities their  values  in  terms  of  the  discharges,  they  become 


from  which  q1  and  q2  are  best  obtained  by  trial ;  although  by 
solution  the  value  of  each  may  be  directly  expressed  by  a 


250  FLOW   THROUGH   PIPES  CHAP.  VIII 

quadratic  equation  in  terms  of  the  given  data,  the  expres- 
sions for  ql  and  q2  are  too  complicated  for  general  use. 
When  it  is  required  to  determine  the  diameters  from  the 
given  lengths,  heads,  and  discharges,  there  are  three  un- 
known quantities,  d,  dlt  d2,  to  be  found  from  only  two  equa- 
tions, and  the  problem  is  indeterminate.  If,  however,  d 
be  assumed,  values  of  dv  and  d2  may  be  found;  and  as  d 
may  be  taken  at  pleasure,  it  appears  that  an  infinite  num- 
ber of  solutions  is  possible.  Another  way  is  to  assume  a 
value  of  y,  corresponding  to  a  proper  pressure -head  at  the 
junction;  then  the  diameters  are  directly  found  from 
formula  (93)  3  for  long  pipes,  in  which  h  is  replaced  by  y 
for  the  large  main,  and  by  H1  —  y  and  H2  —  y  for  the  two 
branches. 

When  two  reservoirs,  Al  and  A2,  are  at  a  higher  elevation 
than  a  third  one  into  which'  they  are  to  deliver  water  by 
pipes  of  lengths  /t  and  /2,  both  of  which  connect  with  a 
third  pipe  of  length  /  which  leads  to  the  third  reservoir,  the 
above  formulas  also  apply.  In  this  case  Hl  and  H2  are 
the  heights  of  the  water  levels  in  the  reservoirs  Al  and  A2 
above  that  in  the  third  reservoir. 

When  the  principal  main  of  a  water-supply  system 
enters  a  town,  it  divides  into  branches  which  deliver  the 
water  to  different  districts,  and  when  such  branches  con- 
nect again  with  the  principal  main  they  form  what  may 
be  called  "diversions."  Fig.  1006  shows  a  simple  case, 
A  being  the  reservoir  and  A B  the  principal  main,  while 
the  pipe  lines  BCE  and  BDE  form  two  routes  or  diversions 
through  which  water  can  flow  to  F.  Let  the  main  AB 
have  the  length  /  and  the  diameter  d,  the  line  BCE  the 
length  /j  and  the  diameter  dv  the  line  BDE  the  length  /2 
and  the  diameter  dv  while  the  line  EF  has  the  length  /3 
and  the  diameter  ds.  Suppose  that  no  water  is  drawn 
from  the  pipes  except  at  F  and  beyond,  that  the  pressure- 
head  Ff  at  F  is  h3,  and  that  the  static  head  F^  on  F  is  h, 
and  let  it  be  required  to  find  the  velocity  and  discharge 


ART.  100  BRANCHES  AND  DIVERSIONS  251 

for  each  of  the  pipes.  The  total  head  H  lost  in  friction  is 
h—hy  and  if  W,  Wv  Wv  and  W3  represent  the  weights  of 
water  that  pass  any  sections  of  the  four  pipes  per  second, 


— 

H 


FIG.  1006 


the  theorem  of  energy,  neglecting  the  entrance  head  at  A 
and  the  velocity-head  at  F,  gives 


Now  referring  to  the  figure  where  piezometers  are  shown 
on  the  profile  at  B  and  E  it  is  seen  that  the  loss  of  head 
in  friction  is  the  same  for  the  diversions  BCE  and  BDE\ 
accordingly  there  must  exist  the  condit'on 


and  since  W  equals  W1  +  Wz  and  also  equals  Wv  the  above 
energy  equation  reduces  to  the  simple  form 


The  values  of  vt  and  va  in  terms  of  T;  are  now  to  be  inserted 
in  this  equation  in  order  to  determine  v.  From  the  con- 
ditions of  continuity  of  flow  and  that  of  equality  of  friction- 
head  in  the  diversions,  are  found  three  equations, 


252  FLOW  THROUGH  PIPES  CHAP,  vin 

and  accordingly,  if  the  square  roots  of  the  quantities  flll/dl 
and  /2/2/d2  be  called  e^  and  ez  for  the  sake  of  abbreviation, 


*  ~  d  «  ~  e,d?  +  e?d 

The  above  formula  for  H  then  reduces  to 


from  which  v  can  be  computed.     Then  vv  vv  and  v3  may. 
be  found,  as  also  the  discharges  q,  qlt  q2,  and  q3. 

As  a  numerical  example,  let  /  =  10  ooo,  /t  =  2200,  /2  =  2800, 
/?  =  i2oo  feet,  and  d  =  i2,  ^=8,  d2  =  io,  <i3  =  io  inches;  let 
F  be  184  feet  below  the  water  level  in  the  reservoir  and 
let  the  required  pressure-head  at  F  be  155  feet,  so  that 
H  =  29  feet.  Taking  for  the  friction  factors  the  mean  value 
0.02  (Art.  86),  the  value  of  fl/d  is  200,  that  of  flljd1  is  66, 
that  of  /2/2/d2  is  67.2,  and  that  of  fj<z/d3  is  28.8.  The  value 
of  e1  is  then  8.12  and  that  of  e2  is  8.20,  while  d/da  is  1.2.  In- 
serting these  in  the  last  formula,  there  is  found  v  =  2.45  feet 
per  second;  then  1^  =  2.16,  v2  =  2.i4,  and  ^3  =  3.53  feet  per 
second.  As  a  check  on  these  results  the  friction-heads  for 
the  four  pipes  may  be  computed,  and  these  are  found  to 
be  1  8.  6  feet  for  /,  4.8  feet  for  /x  and  /2,  and  5.5  feet  for  Z3; 
the  sum  of  these  is  28.9  feet,  which  is  a  sufficiently  close 
agreement  with  the  given  29.0  feet  for  a  preliminary  com- 
putation. The  discharges  are  q  =  q.A  =  i  .93,  ql  =  o.  7  5  ,  q2  =  i  .  1  8 
cubic  feet  per  second,  and  the  sum  of  q1  and  q2  equals  q, 
as  should  be  the  case.  The  computation  may  now  be 
repeated,  if  thought  necessary,  the  above  velocities  being 
used  to  take  better  values  of  the  friction  factors  from 
Table  33. 

There  are  marked  analogies  between  the  flow  of  water 
in  pipes  and  the  flow  of  electricity  in  metallic  conductors. 
Thus  in  Fig.  1006,  let  BCE  and  BDE  be  two  wires  that 


ART.  100  BRANCHES  AND  DIVERSIONS  253 

carry  the  electric  current  passing  from  A  to  F.  If  Cl  and 
C2  be  the  currents  in  these  circuits  and  R^  and  jR2  the  resist- 
ances of  the  wires,  it  is  an  electric  law  that  R1Cl=R2C2f 
or  the  currents  are  inversely  as  the  resistances.  For  water 
the  discharges  ql  and  q2  are  analogous  to  the  electric  cur- 
rents, and,  from  the  above  equation  which  expresses  the 
equality  of  the  friction-heads,  it  is  seen  that 


and  accordingly  the  same  law  holds  if  the  coefficients  of 
q^  and  q2  be  called  resistances.  If  there  be  a  third  diver- 
sion BGE  of  length  /4  and  diameter  d4  connecting  B  and  E, 
the  current  or  the  discharge  through  AB  divides  between 
the  three  diversions  according  to  the  same  law,  and 


from  which  it  is  seen  that  (fJ4/d45)?q4  is  equal  to  each  of  the 
corresponding  expressions  for  the  other  diversions.  This 
subject  will  receive  further  discussion  in  Art.  193. 

Prob.  lOOa.  In  Fig.  lOOa  let  ql  =  o-S  and  <?2  =  o.4  cubic  feet 
per  second;  Hl  =  140  and  H2=  125  feet;  ^  =  3810,  ^  =  2455,  and 
I=i2  314  feet.  If  dl  equals  d2  find  the  values  of  d  and  dlt  and 
also  the  pressure-head  at  the  junction  if  its  depth  below  the 
reservoir  level  is  108  feet. 

Prob.  1006.  Connecting  two  points  M  and  N  there  are  three 
diversions  having  the  lengths  /lf  /3,  /3  and  the  diameters  dlt  d2,  d3. 
If  q  be  the  total  discharge  of  the  three  pipes,  show  that 


in  which  D  represents  the  quantity  ^3J12+^1^3af22  +  ^2<i32,  while 
eltez,  e3  are  the  square  roots  of  fA/d/,  /2/2/^25  * 


Prob.  lOOc.  From  a  reservoir  A  a  pipe  10  ooo  feet  long  and 
1  6  inches  in  diameter  runs  to  a  point  B  from  which  two  diver- 
sions lead  to  E.  The  diversion  BCE  is  1600  feet  long  and  10 
inches  in  diameter,  while  BDE  consists  of  2000  feet  of  lo-inch 
pipe  and  1  500  feet  of  8-inch  pipe.  From  the  junction  E,  a  pipe 


254 


FLOW   THROUGH    PlPES 


CHAP.  VIII 


EF,  1000  feet  long  and  12  inches  in  diameter,  leads  to  the  busi- 
ness section  of  the  town,  where  it  is  desired  to  have  four  fire 
streams  deliver  a  total  discharge  of  900  gallons  per  minute 
through  four  hose  lines  of  2^-inch  smooth  rubber-lined  hose 
and  ij-inch  smooth  nozzles.  The  point  F  is  180  feet  below 
the  water  level  in  the  reservoir.  Compute  the  velocity  and  dis- 
charge for  each  pipe  and  hose  line,  the  pressure-head  at  -F,  and 
the  friction-head  lost  in  each  pipe  and  hose  line. 


ART.  101.     RIVETED  AND  WOOD  PIPES 

Large  pipes  are  sometimes  made  /oi  wrought  iron  or 
steel  plates  riveted  together.  Each  section  usually  con- 
sists of  a  single  plate  which  is  bent  into  the  circular  form 
and  the  edges  united  by  a  longitudinal  riveted  lap  joint. 
The  different  sections  are  then  riveted  together  in  trans- 
verse joints  so  as  to  form  a  continuous  pipe.  At  A B 


eTTT!  ; .  oo/ 

1-— -fr 


t 


FIG.  i 01 

is  shown  the  so-called  taper  joint,  where  the  end  of  each 
section  goes  into  the  end  of  the  following  one,  as  in  a 
stove-pipe,  the  flow  occurring  in  the  direction  from  A 
to  B.  At  CD  is  seen  the  method  of  cylinder  joints  where 
the  sections  are  alternately  larger  and  smaller.  For  the 
large  sizes  double  rows  of  rivets  are  used  both  in  the 
longitudinal  and  transverse  joints.  Riveted  pipes  have 
also  been  built  with  butt  transverse  joints,  a  lap  plate 
being  used  on  the  outside. 

Pipes  of  this  kind  have  long  been  in  use  in  California 
in  temporary  mining  operations,  the  diameters  being 
from  0.5  to  1.5  feet.  In  1876  one  was  laid  at  Rochester, 
N.  Y.,  partly  2  and  partly  3  feet  in  diameter.  Since  1892 
several  lines  of  large  diameter  have  been  constructed, 
notably  the  East  Jersey  pipe  of  3,  3.5,  and  4  feet  diameter, 


ART.  101  RIVETED  AND  WOOD  PIPES  255 

the  Allegheny  pipe  of  5  feet  diameter,  and  the  Ogden  pipe 
of  6  feet  diameter. 

Owing  to  the  friction  of  the  rivets  and  joints  the  dis- 
charge of  such  riveted  pipes  is  less  than  that  of  common 
cast-iron  pipes.  The  following  values  of  the  friction 
factor  /,  which  have  been  derived  from  the  data  given 
by  Herschel,*  are  applicable  to  new  clean  riveted  pipes, 
coated  in  the  usual  manner  with  asphaltum: 

Velocity,  feet  per  second,  v=    i  2  3  4  5  6 

T   .    ^  (     T,  ft.  diam.,     /  =0.03  5     0.029     0.024     0.021     0.019     0.017 
Cylinder  Joints]    ^  ft   diam ;     /=0.O25     0.022     0.026     0.020     0.021     0.021 

T         .j  .  j  3$  ft.  diam.,     /  =  0.027     0.023     0.022     0.021     0.021     0.022 

I    4ft.  diam.,    /  =0.027     0.026     0.025     0.024     0.023     °-O23 

These  friction  factors  are  approximately  double  the  values 
given  for  new  cast-iron  pipes  in  Table  33,  this  increase 
being  mostly  due  to  the  friction  of  the  rivet  heads.  It 
should  be  noted  that  these  friction  factors  increase  with 
age,  Herschel' s  gagings  showing  that  after  four  years'  use 
the  cylinder  pipe  of  4  feet  diameter  gave  /  =  o.o34  for 
v  =  i,  /  =  0.028  for  v  =  2,  and  /  =  0.026  for  v  greater  than 
2  feet  per  second.  In  designing  a  pipe  an  allowance 
should  be  made  for  this  fact. 

Gagings  by  Marx,  Wing,  and  Hoskinsf  of  the  flow 
through  a  steel-riveted  pipe  6  feet  in  diameter  with  butt 
joints,  when  new  and  again  after  two  years'  use,  furnish 
the  following  values  of  the  friction  factor  /  corresponding 
to  several  velocities  in  feet  per  second : 

Velocity,  v=     i  2  3  4  5 

1897,  /  =O.O2 1      O.O2I      O.O22      O.O2I      

1899,        /  =0.039    0.027    0.025    0.024    0.023 

These  also  show  that  the  roughness  of  the  surface  materially 
increases  with  time. 

*  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 
Conduits.     New  York,  1897. 

f  Transactions   American    Society   of   Civil   Engineers,    1898,    vol.    40, 
p.  471;  and  1900,  vol.  44,  p.  34. 


256  FLOW  THROUGH  PIPES  CHAP,  vm 

Wood  pipes  were  used  in  several  American  cities  during 
the  years  1750-1850,  these  being  made  of  logs  laid  end 
to  end,  a  hole  3  or  4  inches  in  diameter  having  been  first 
bored  through  each  log.  Pipes  formed  of  redwood  staves 
were  first  used  in  California  about  1880,  these  staves 
being  held  in  place  by  bands  of  wrought  iron  arranged 
so  that  they  could  be  tightened  by  a  nut  and  screw.  Sev- 
eral long  lines  of  these  large  conduit  pipes  have  been 
built  in  the  Rocky  mountains  and  Pacific  states,  and 
they  have  also  been  used  there  for  city  water  mains  to  a 
limited  extent. 

Gagings  of  a  wood  pipe  6  feet  in  diameter  were  made 
by  Marx,  Wing,  and  Hoskins  in  connection  with  those  of 
the  steel  pipe  cited  above.  The  values  of  the  friction 
factor  /  deduced  from  their  results  for  velocities  ranging 
from  i  to  5  feet  per  second  are  as  follows: 

Velocity,  v=     i  2  3  4  5 

1897,  /=o.O26         0.019         0.017         0.016         

1899,  /=o.oi9         0.018         0.017      k  0.017         0.017 

These  show  that  this  wood   pipe   became    smoother   after 
two  years'  use,  while  the  steel  pipe  became  rougher. 

Noble's  gagings  of  wood  pipes  3.67  and  4.51  feet  in 
diameter  furnish  similar  values  of  /.*  For  the  smaller 
pipe  /  ranges  from  0.021  to  0.019,  with  velocities  ranging 
from  3.5  to  4.8  feet  per  second.  For  the  larger  pipe  / 
ranges  from  0.019  to  0.016,  with  velocities  ranging  from 
2.3  to  4.7  feet  per  second.  -From  Adams'  measurements 
on  a  pipe  1.17  feet  in  diameter  the  values  of  /  range  from 
0.027  to  0.02,  with  velocities  ranging  from  0.7  to  1.5  feet 
per  second.  Noble's  discussion  of  all  the  recorded  gagings 
on  wood-  pipes  show  certain  unexplained  discrepancies, 
and  he  proposes  special  empirical  formulas  to  be  used 
for  precise  computations.  Wooden  stave  pipes  after 

*  Transactions   American     Society   of    Civil    Engineers,    1902,    vol.  49, 
pp.  112,  143. 


ART.  102  FIRE  HOSE  257 

being  in  service  some  time  may  undergo  considerable 
alteration  in  form,  as  the  circle  is  apt  to  be  deformed 
into  an  ellipse. 

By  the  help  of  the  formulas  of  the  preceding  pages, 
computations  for  the  velocity  and  discharge  of  steel  and 
wood  pipes  under  given  heads  may  be  readily  made. 
As  such  pipes  are  generally  long  the  formulas  of  Art.  93 
will  usually  apply.  In  designing  a  steel  pipe  a  liberal 
factor  of  safety  should  be  introduced  by  taking  a  value 
of  /  sufficiently  large  so  that  the  discharge  may  not  be 
found  deficient  after  a  few  years'  use  has  deteriorated 
its  surface. 

Prob.  lOla.  Find  the  diameter  of  a  riveted  pipe,  ten  miles 
long,  to  deliver  30  million  gallons  per  day  under. a  head  of  105.6 
feet. 

Prob.  1016.  What  is  the  discharge,  in  gallons  per  day,  of  a 
wood  stave  pipe  5  feet  in  diameter  when  the  slope  of  the  hy- 
draulic gradient  is  47.5  feet  per  mile? 

ART.  102.     FIRE  HOSE 

Fire  hose  is  generally  2\  inches  in  diameter,  and  lined 
with  rubber  to  reduce  the  frictional  losses.  The  following 
values  of  the  friction  factor  /  have  been  deduced  from 
the  experiments  of  Freeman.* 

Velocity  in  feet  per  second,      o>=     4            6  10  15          20 

Unlined  linen  hose,                     /  =0.038  0.038  0.037  0.035  0.034 

Rough  rubber-lined  cotton,      /  =0.030  0.031  0.031  0.030  0.029 

Smooth  rubber-lined  cotton,    /  =0.024  0.023  0.022  0.019  0.018 

Discharge,  gallons  per  minute    =     61           92  153         230        306 

By  the  help  of  this  table  computations  may  be  made 
on  flow  of  water  through  fire  hose,  in  the  same  manner  as 
for  pipes.  It  is  seen  that  the  friction  factors  for  the  best 
hose  are  slightly  less  than  those  given  for  2j-inch  pipes 
in  Table  33. 

*  Transactions   American    Society    of    Civil   Engineers,    1889,    vol.   21, 
PP-  303,  346. 


258  FLOW  THROUGH  PIPES  CHAP,  vin 

When  the  hose  line  runs  from  a  steamer  to  the  nozzle, 
instead  of  from  a  reservoir,  the  head  h  is  that  due  to  the 
pressure  p  at  the  steamer  pump  (Art.  11).  If  this  hose 
line  is  of  uniform  diameter  the  velocity  in  the  hose  and 
nozzle  may  be  computed  by  Art.  97  and  the  discharge  is 
then  readily  found.  For  example,  let  the  hose  be  i\ 
inches  in  diameter  and  400  feet  long,  the  pressure  at  the 
steamer  be  100  pounds  per  square  inch,  which  corresponds 
to  a  head  of  230.4  feet,  and  the  nozzle  be  ij  inches  in 
diameter  with  a  coefficient  of  velocity  of  0.98.  Then, 
neglecting  the  loss  of  head  at  entrance,  and  using  for  / 
the  value  0.03,  the  velocity  from  the  nozzle  is  found  to 
be  66.0  feet  per  second,  which  gives  a  velocity-head  of 
67.7  feet  and  a  discharge  of  180  gallons  per  minute.  The 
head  lost  in  friction  is  230.4  —  67.7  =  162.7  feet,  of  which 
2.8  feet  is  lost  in  the  nozzle  and  the  remainder  in  the  hose. 

.  Sometimes  the  hose  near  the  steamer  is  larger  in  diam- 
eter than  the  remaining  length.  Let  ^  be  the  length  and 
dj  the  diameter  of  the  larger  hose,  and  12  and  d2  the  same 
quantities  for  the  smaller  hose.  Let  c}  be  the  coefficient 
of  velocity  for  a  smooth  nozzle,  D  its  diameter,  and  V  the 
velocity  of  the  stream  issuing  from  the  nozzle.  By  reason- 
ing as  in  Arts.  89  and  97,  and  neglecting  losses  of  head  at 
entrance  and  in  curvature,  there  is  found 


V- 


and  the  discharge  is  given  by  q  =  \nD2V.  For  example, 
let  h  =  230.4,  Zj  =  100,  /2  =  300  feet;  dt  =3,  d2  =  2.5,  D  =  1.125 
inches;  ^=0.98,  and  ^=f  2=0.03.  Then,  by' the  formula, 
1^  =  69.7  feet  per  second,  which  gives  a  velocity-head  of 
75.5  feet  and  a  discharge  of  190  gallons  per  minute.  This 
example  is  the  same  as  that  of  the  preceding  paragraph, 
except  that  a  larger  hose  is  used  for  one-fourth  of  the  length, 


ART.  102  FIRE  HOSE  259 

and  it  is  seen  that  its  effect  is  to  increase  the  velocity-head 
nearly  12  percent  and  the  discharge  nearly  6  percent.  For 
this  case  the  head  lost  in  friction  is  154.9  feet,  of  which  3.1 
feet  is  lost  in  the  nozzle  and  the  remainder  in  the  hose. 

In  using  the  above  formula  the  tip  of  the  nozzle  is  sup- 
posed to  be  on  the  same  level  with  the  pressure  gage  at 
the  steamer  pump  and  the  head  h  is  given  in  feet  by  2.304^, 
^here  p  be  the  gage  reading  in  pounds  per  square  inch. 
If  the  tip  of  the  nozzle  is  a  vertical  distance  z  above  this 
gage,  h  is  to  be  replaced  by  h  —  z  in  the  formula;  if  it  be 
the  same  vertical  distance  below  the  gage,  h  is  to  be  re- 
placed by  h  +  z.  In  the  former  case  gravity  decreases  and 
in  the  latter  case  it  increases  the  velocity  and  discharge. 
The  above  formula  applies  also  to  the  case  of  a  hose  con- 
nected to  a  hydrant,  if  h  be  the  effective-head  at  the  en- 
trance, that  is,  the  pressure-head  plus  the  velocity-head 
in  the  hydrant.  In  Art.  192  will  be  found  further  dis- 
cussions regarding  pumping  through  fire  hose. 

Prob.  102a.  For  the  above  numerical  examples  compute  the 
head  lost  in  the  hose  and  that  lost  in  the  nozzle. 

Prob.  1026.  When  the  pressure  gage  at  the  steamer  indi- 
cates 83  pounds  per  square  inch,  a  gage  on  the  leather  hose  800 
feet  distant  reads  25  pounds.  Compute  the  value  of  the  friction 
factor  /,  the  discharge  per  minute  being  121  gallons.  If  the 
second  gage  be  at  the  entrance  to  a  i^-inch  nozzle,  compute  its 
coefficient  of  velocity. 

Prob.  102c.  At  a  hydrant  of  diameter  dl  the  pressure-head 
is  hr  To  this  is  attached  a  hose  of  length  /  and  diameter  d± 
and  to  the  end  of  the  hose  a  nozzle  of  diameter  D  and  velocity 
coefficient  cv  Neglecting  losses  at  entrance  and  in  curvature 


is  the  formula  for  computing  the  velocity  of  the  jet  issuing  from 
the  nozzle  when  its  tip  is  held  at  the  same  level  as  the  gage  that 
indicates  the  pressure-head. 


260  FLOW  THROUGH  PIPES  CHAP,  viii 


ART.  103.     OTHER  FORMULAS  FOR  FLOW  IN  PIPES 

The  formulas  thus  far  presented  in  this  chapter  are 
based  upon  the  assumption  that  all  losses  of  head  vary 
with  the  square  of  the  velocity.  This  is  closely  the  case 
for  the  velocities  common  in  engineering  practice,  but  for 
velocities  smaller  than  0.5  feet  per  second  the  losses  of 
head  due  to  friction  have  been  found  to  vary  at  a  less 
rapid  rate,  and  in  fact  nearly  as  the  first  power  of  the 
velocity.  Probably  at  usual  velocities  the  loss  of  head  in 
friction  is  composed  of  two  parts,  a  small  part  varying 
directly  with  the  velocity  which  is  due  to  cohesive  resist- 
ance along  the  surface,  and  a  large  part  varying  as  the 
square  of  the  velocity  which  is  due  to  impact  as  illustrated 
in  Fig.  86.  This  was  recognized  by  the  early  hydraulicians 
who,  after  denning  the  friction-head  and  friction-factor 
as  in  (86),  by  the  formula 


endeavored  to  express  /  in  terms  of  the  velocity  v.  Thus, 
D  'Aubisson  deduced 

0.00484 

/=0.0269  + f~ 

and  Weisbach  advocated  the  form 

0.00172 
7  =  0.0144  H 7=- 

Vv 

Darcy,  on  the  other  hand,  expressed  /  in  terms  of  d,  namely,, 

0.00167 
/  =  0.0199  H -j — - 

All  these  expressions  are  for  English  measures,  v  being  in 
feet  per  second  and  d  in  feet.  Later  investigations  show, 
however,  that  /  varies  with  both  v  and  d,  and  the  best  that 
can  now  be  done  is  to  tabulate  its  values  as  in  Table  33. 
In  fact  it  may  be  said  that  the  theory  of  the  flow  of  water 
in  pipes  at  common  velocities  is  not  yet  well  understood. 


ART.  103       OTHER  FORMULAS  FOR  FLOW  IN  PIPES  261 

Many  attempts  have  been  made  to  express  the  velocity 
of  flow  in  a  long  pipe  by  an  equation  of  the  form 


in  which  a,  /?,  and  7-  are  to  be  determined  from  experiments 
in  which  v,  d,  h,  and  /  have  been  measured.  The  ex- 
ponential formula  deduced  by  Lampe  for  clean  cast-iron 
pipes  varying  in  diameter  from  one  to  two  feet  is 


55  (103) 

in  which  d,  h,  and  /  are  to  be  taken  in  feet,  and  v  will  be 
found  in  feet  per  second.     From  this  are  derived 


by  which  discharge  and  diameter  may  be  computed.  Other 
investigators  find  different  values  of  ft  and  /-,  the  values 
/?  =  §  and  f  =  \  being  frequently  advocated. 

The  formula  of  Ckezy  (Art.  106),  that  of  Kutter  (Art. 
Ill),  and  that  of  Bazin  (Art.  115)  are  frequently  used  for 
the  discussion  of  long  pipes,  care  being  taken  to  select  the 
proper  value  of  c  for  the  first,  of  n  for  the  second,  and  of 
r>n  for  the  third.  In  some  cases  the  use  of  the  formulas  of 
Kutter  and  Bazin  is  more  advantageous  than  those  of  the 
preceding  cases,  because  they  enable  the  influence  of  the 
roughness  of  the  surface  to  be  better  taken  into  account. 

The  formulas  of  this  chapter  do  not  apply  to  very  small 
pipes  and  very  low  velocities,  and  it  is  well  known  that  for 
such  conditions  the  loss  of  head  in  friction  varies  as  the 
first  power  of  the  velocity.  This  was  shown  in  1843  by 
Poiseuille  who  made  experiments  in  order  to  study  the 
phenomena  of  the  flow  of  blood  in  veins  and  arteries.  For 
pipes  of  less  than  0.03  inches  diameter  he  found  the  head  h 
to  be  given  by  h  =  CJv/d2,  where  Cl  is  a  constant  factor  for  a 
given  temperature,  v  is  the  velocity,  d  the  diameter,  and  I 
the  length  of  the  pipe.  Later  researches  indicate  that  the 


262  FLOW  THROUGH  PIPES  CHAP,  vin 

laws  expressed  by  this  equation  also  hold  for  large  pipes 
provided  the  velocity  be  very  small,  and  that  there  is  a 
certain  critical  velocity  at  which  the  law  changes  and  beyond 
which  h=C2lv2/d,  as  for  the  common  cases  in  engineering 
practice.  This  critical  point  appears  to  be  that  where 
the  filaments  cease  to  move  in  parallel  lines  and  where  the 
impact  disturbances  illustrated  in  Fig.  86  begin.  For  a 
very  small  pipe  the  velocity  may  be  high  before  this  critical 
point  is  reached;  for  a  large  pipe  it  happens  at  very  low 
velocities.  Experiments  devised  by  Reynolds  enable  the 
impact  disturbance  to  be  actually  seen  as  the  critical  veloc- 
ity is  passed,  so  that  its  existence  is  beyond  question.  It 
may  also  be  noted  that  the  velocity  of  flow  through  a  sub- 
merged sand  filter  bed  varies  directly  as  the  first  power  of 
the  effective  head. 

Prob.  103.  Solve  Problems  90  and  91  by  the  use  of  the  above 
formulas  of  Lampe. 


ART.  104.     COMPUTATIONS  IN  METRIC  MEASURES 

Nearly  all  the  formulas  of  this  chapter  are  rational  in 
form,  the  coefficient  of  velocity  cv  the  factors  /  and  fv  and 
the  factors  m,  mv  m2,  and  m'  are  abstract  numbers  and  may 
be  used  in  any  system  of  measures. 

(Art.  86)  The  mean  value  of  the  friction  factor  /  is 
0.02,  and  Table  34  gives  closer  values  corresponding  to 
metric  arguments.  For  example,  let  /  =  3ooo  meters, 
^  =  30  centimeters  =0.3  meters,  and  ^  =  1.75  meters  per 
second.  Then  from  the  table  /is  0.022,  and 


h"  =0.022  X~       X-^f-  =34-3  meters 

which  is  the  probable  loss  of  head  in  friction.  By  the  use 
of  Table  36  approximate  computations  may  be  made  more 
rapidly;  thus  for  this  case  the  loss  of  head  for  100  meters 


ART.  104         COMPUTATIONS  IN  METRIC  MEASURES  263 

/ 

of  pipe  is  found  to  be  i.io  meters,  hence  for  3000  meters 
the  loss  of  head  is  33  meters. 

(Art.  90)  The  metric  value  of  \K\/  zg  is  3.477  and 
that  of  8/Vg  is  0.2653. 

(Art.  91)  When  (91)  is  used  in  the  metric  system  the 
constant  0.4789  is  to  be  replaced  by  0.6075;  nere  <?  is  to 
ba  in  cubic  meters  per  second,  and  /  and  d  in  meters. 

(Art.  93)  In  (93)2  the  two  constants  are  4.43  and 
3.48  instead  of  8.02  and  6.30.  In  (93)  3  the  constant  is 
0.607  instead  of  0.479. 

(Art.  101)  The  friction  factors  /  for  steel  and  wood 
pipes  may  be  taken  for  metric  arguments  by  using  the 
velocities  in  meters  per  second,  namely,  by  writing  0.3, 
0.6,  0.9,  1.2,  1.5,  1.8  meters  per  second,  instead  of  i,  2, 
3,  4,  5,  6  feet  per  second. 

(Art.  102)  For  fire  hose  the  values  of  the  friction 
factor  /  for  metric  data  are  as  follows,  for  hose  6.35 
centimeters  in  interior  diameter: 

Velocity,  meters  per  second,  v  =  i.  2  2  1.83  3.05  4.57  6.10 

Unlined  linen  hose,                    /  =0.038  0.038  0.037  °-°35  °-°34 

Rough  rubber-lined  cotton,     /  =0.030  0.031  0.031  0.030  0.029 

Smooth  rubber-lined  cotton,  /  =0.024  0.023  0.022  0.019  0.018 

Discharge,  liters  per  minute,             193  348  579  871  1158 

(Art.  103)  In  the  metric  system  the  formulas  for 
the  friction  factor  /  are  the  same  as  those  in  the  text, 
except  that  the  numerator  of  the  last  term  is  to  be  divided 
by  3.28  in  the  formulas  of  D'Aubisson  and  Darcy  and 
by  i.  8  1  in  that  of  Weisbach.  Lampe's  formula  is 


and  his  formulas  for  discharge  and  diameter  are 


in  which  d,  h,  and  /  are  in  meters,  v  in  meters  per  second, 
and  q  in  cubic  meters  per  second. 


264  FLOW  THROUGH  PIPES  CHAP,  viir 

Prob.  104a.  Compute  the  diameter,  in  centimeters,  for  a 
pipe  to  deliver  500  liters  per  minute  under  a  head  of  2  meters, 
when  its  length  is  100  meters.  Also  when  the  length  is  1000 
meters. 

Prob.  1046.  Compute  the  velocity-head  and  discharge  for  a 
pipe  i  meter  in  diameter  and  856  meters  long  under  a  head  of 
64  meters.  Compute  the  same  quantities  when  a  smooth  nozzle 
5  centimeters  in  diameter  is  attached  to  the  end  of  the  pipe. 

Prob.  104:0.  A  compound  pipe  has  the  three  diameters  15, 
20,  and  30  centimeters,  the  lengths  of  which  are  150,  600,  and 
430  meters.  Compute  the  discharge  under  a  head  of  16  meters. 

Prob.  104J.  A  steel  riveted  pipe  1.5  meters  in  diameter  is 
7500  meters  long.  Compute  the  velocity  and  discharge  under 
a  head  of  30.5  meters. 

Prob.  104<?.  The  value  of  Cl  in  Poiseuille's  formula  for  small 
pipes  is  0.0000177  f°r  English  measures  at  10°  centigrade.  Show 
that  its  value  is  0.0000690  for  metric  measures. 

Prob.  104/.  In  Fig.  1006  let  the  pipe  A B  be  3000  meters  long 
and  30  centimeters  in  diameter,  BCD  be  800  feet  long  and  20 
centimeters  in  diameter,  BDE  be  1000  feet  long  and  20  centi- 
meters in  diameter,  and  EF  be  300  meters  long  and  30  centi- 
meters in  diameter.  Compute  the  velocity  and  discharge  for 
each  pipe  when  the  total  lost  head  H  is  12.5  meters. 


ART.  105  DEFINITIONS  265 


CHAPTER  IX 
FLOW  IN  CONDUITS  AND  CANALS 

ART.  105.     DEFINITIONS 

From  the  earliest  times  water  has  been  conveyed 
from  place  to  place  in  artificial  channels,  such  as  troughs, 
aqueducts,  ditches,  and  canals,  there  being  no  head  to 
cause  the  flow  except  that  due  to  the  slope.  The  Roman 
aqueducts  were  usually  rectangular  channels  about  2J- 
feet  wide  and  5  feet  deep,  lined  with  cement,  sometimes 
running  underground  and  sometimes  supported  on  arches. 
The  word  ' '  conduit ' '  will  be  used  as  a  general  term  for 
a  channel  of  any  shape  lined  with  timber,  mortar,  or 
masonry,  and  will  also  include  large  metal  pipes,  troughs, 
and  sewers.  Conduits  may  be  either  open  as  in  the  case 
of  troughs,  or  closed  as  in  sewers  and  most  aqueducts. 
Ditches  and  canals  are  conduits  in  earth  without  artificial 
lining.  Most  of  the  principles  relating  to  conduits  and 
canals  apply  also  to  streams,  and  the  word  channel  will 
be  used  as  applicable  to  all  cases. 

The  wetted  perimeter  of  the  cross-section  of  a  channel 
is  that  part  of  its  boundary  which  is  in  contact  with  the 
water.  Thus,  if  a  circular  sewer  of  diameter  d  be  half 
full  of  water  the  wetted  perimeter  is  \nd.  In  this  chapter 
the  letter  p  will  designate  the  wetted  perimeter. 

The  hydraulic  radius  of  a  water  cross-section  is  its 
area  divided  by  its  wetted  perimeter,  and  the  letter  r 
will  be  used  to  designate  it.  If  a  be  the  area  of  the  cross- 
section,  the  hydraulic  radius  is  found  by 

r^a/p 


266  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

The  letter  r  is  of  frequent  occurrence  in  formulas  for  the 
flow  in  channels;  it  is  a  linear  quantity  which  is  always 

expressed  in  the  same 
unit  as  p  and  hence  its 
numerical  value  is  dif- 
ferent in  different  sys- 
tems of  measures.  It  is  frequently  called  the  hydraulic 
depth  or  hydraulic  mean  depth,  because  for  a  shallow 
section  its  value  is  but  little  less  than  the  mean  depth 
of  the  water.  Thus  in  Fig.  105,  if  b  be  the  breadth  on 
the  water  surface,  the  mean  depth  is  a/ b,  and  the  hydraulic 
radius  is  a/p\  and  these  are  nearly  equal,  since  p  is  but 
slightly  larger  than  b. 

The  hydraulic  radius  of  a  circular  cross-section  filled 
with  water  is  one-fourth  of  the  diameter;  thus 

The  same  value  is  also  applicable  to  a  circular  section 
half  filled  with  water,  since  then  both  area  and  wetted 
perimeter  are  one-half  their  former  values. 

The  slope  of  the  water  surface  in  the  longitudinal 
section,  designated  by  the  letter  s,  is  the  ratio  of  the  fall 
h  to  the  length  /  in  which  that  fall  occurs,  or 

s-k/l 

The  slope  is  hence  expressed  as  an  abstract  number,  which 
is  independent  of  the  system  of  measures  employed. 
To  determine  its  value  with  precision  h  must  be  obtained 
by  referring  the  water  level  at  each  end  of  the  line  to  a 
bench-mark  by  the  help  of  a  hook  gage  or  other  accurate 
means,  the  benches  being  connected  by  level  lines  run 
with  care.  The  distance  /  is  measured  along  the  inclined 
channel,  and  it  should  be  of  considerable  length  in  order  that 
the  relative  error  in  h  may  not  be  large.  If  s=o  there 
Is  no  slope  and  no  flow;  but  if  there  be  even  the  smallest 


ART.  105  DEFINITIONS  267 

slope  the  force  of  gravity  furnishes  a  component  acting 
down  the  inclined  surface,  and  motion  ensues.  The  ve- 
locity of  flow  evidently  increases  with  the  slope. 

The  flow  in  a  channel  is  said  to  be  steady  when  the 
same  quantity  of  water  per  second  passes  through  each 
cross-section.  If  an  empty  channel  be  filled  by  admitting 
water  at  its  upper  end  the  flow  is  at  first  non-steady  or 
variable,  for  more  water  passes  through  one  of  the  upper 
sections  per  second  than  is  delivered  at  the  lower  end. 
But  after  sufficient  time  has  elapsed  the  flow  becomes 
steady;  when  this  occurs  the  mean  velocities  in  different 
sections  are  inversely  as  their  areas  (Art.  32). 

Uniform  flow  is  that  particular  case  of  steady  flow 
where  all  the  water  cross-sections  are  equal,  and  the 
slope  of  the  water  surface  is  parallel  to  that  of  the  bed 
of  the  channel.  If  the  sections  vary  the  flow  is  said  to 
be  non-uniform,  although  the  condition  of  steady  flow 
is  still  fulfilled.  In  this  chapter  only  the  case  of  uniform 
flow  will  be  discussed. 

The  velocities  of  different  filaments  in  a  channel  are 
not  equal,  as  those  near  the  wetted  perimeter  move  slower 
than  the  central  ones  owing  to  the  retarding  influence 
of  friction.  The  mean  of  all  the  velocities  of  all  the  fila- 
ments in  a  cross-section  is  called  the  mean  velocity  v. 
Thus  if  i/9  v",  etc.,  be  velocities  of  different  filaments, 

v*+  v"+etc. 

v  = • 

n 

in  which  n  is  the  number  of  filaments.  Let  a  be  the  area 
of  the  cross-section  and  let  each  filament  have  the  small 
cross-section  of  area  a';  then  n  =  a/a',  and  hence 

av=*a'(v'+  ?;"+ etc.) 

.But  the  second  member  is  the  discharge  q,  that  is,  the 
quantity  of  water  passing  the  given  cross -section  in  one 


268  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

second.     Therefore   the   mean   velocity   may  be   also   de- 
termined b     the  relation 


The  filaments  which  are  here  considered  are  in  part  imagi- 
nary, for  experiments  show  that  there  is  a  constant  sinuous 
motion  of  particles  from  one  side  of  the  channel  to  the 
other.  The  best  definition  for  mean  velocity  hence  is, 
that  it  is  a  velocity  which  multiplied  by  the  area  of  the 
cross-section  gives  the  discharge,  or  v=q/a. 

Prob.  105a.  Compute  the  hydraulic  radius  of  a  rectangular 
trough  whose  width  is  4.4  feet  and  depth  1.3  feet. 

Prob.  1056.  Compute  the  mean  velocity  in  a  circular  sewer 
of  4  feet  diameter  when  it  is  half  filled  and  discharges  120  gal- 
lons per  second. 

ART.  106.     FORMULA  FOR  MEAN  VELOCITY 

When  all  the  wetted  cross-sections  of  a  channel  are 
equal,  and  the  water  is  neither  rising  nor  falling,  having 
attained  the  condition  of  steady  flow,  the  flow  is  said  to  be 
uniform.  This  is  the  case  in  a  conduit  or  canal  of  constant 
size  and  slope  whose  supply  does  not  vary.  The  same 
quantity  of  water  per  second  then  passes  each  cross-section, 
and  consequently  the  mean  velocity  in  each  section  is 
the  same.  This  uniformity  of  flow  is  due  to  the  resistances 
along  the  interior  surface  of  the  channel,  for  were  it  per- 
fectly smooth  the  force  of  gravity  would  cause  the  veloc- 
ity to  be  accelerated.  The  entire  energy  of  the  water  due 
to  the  fall  h  is  hence  expended  in  overcoming  resistances 
caused  by  surface  roughness.  A  part  overcomes  friction 
along  the  surface,  but  most  of  it  is  expended  in  eddies 
of  the  water,  whereby  impact  results  and  heat  is  generated. 
A  complete  theoretic  analysis  of  this  complex  case  has 
not  been  perfected,  but  if  the  velocity  be  not  small  the 
discussion  given  for  pipes  in  Art.  86  applies  equally  well- 
to  channels. 


ART.  103  FORMULA  FOR  MEAN  VELOCITY  269 

Let  W  be  the  weight  of  water  passing  any  cross-section 
in  one  second,  F  the  force  of  friction  per  square  unit  along 
the  surface,  p  the  wetted  perimeter,  and  h  the  fall  in  the 
length  /.  The  potential  energy  of  the  fall  is  Wh.  The 
total  resisting  friction  is  Fpl,  and  the  energy  consumed  per 
second  is  Fplv,  if  v  be  the  velocity.  Accordingly  Fplv 
equals  Wh.  But  the  value  of  W  is  wav,  if  w  be  the  weight 
of  a  cubic  foot  of  water  and  a  be  the  area  of  the  cross-section 
in  square  feet.  Therefore  Fpl=wah,  and  since  a/p  is  the 
hydraulic  radius  r,  and  h/l  is  the  slope  s,  this  reduces  to 
F  =  wrs,  which  is  an  approximate  expression  for  the  resist- 
ing force  of  friction  on  one  square  unit  of  the  surface  of 
the  channel.  In  order  to  establish  a  formula  for  the  mean 
velocity  the  value  of  F  must  be  expressed  in  terms  of  v, 
and  this  can  only  be  done  by  studying  the  results  of  experi- 
ments. These  indicate  that  F  is  approximately  propor- 
tional to  the  square  of  the  mean  velocity.  Therefore  if 
c  be  a  constant,  the  mean  velocity  is 

v=cVrs  (106) 

which  is  the  formula  first  advocated  by  Chezy  in  1775. 
This  is  really  an  empirical  expression,  since  the  relation 
between  F  and  v  is  derived  from  experiments.  The  coeffi- 
cient c  varies  with  the  roughness  of  the  bed  and  with  other 
circumstances. 

Another  method  of  establishing  Chezy  's  formula  for 
channels  is  to  consider  that  when  a  pipe  on  a  uniform 
slope  is  not  under  pressure,  the  hydraulic  gradient  coincides 
with  the  water  surface.  Then  formula  (86)  may  be  used 
by  replacing  h"  by  h  and  d  by  its  value  41-.  Accordingly 


=--        or 


in  which  VSg/f  is  the  Chezy  coefficient. 

This  coefficient   c   is   different   in   different   systems   of 
measures  since  it  depends  upon  g.     For  the  English  system 


270  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

it  is  found  that  c  usually  lies  between  30  and  160,  and  that 
its  value  varies  with  the  hydraulic  radius  and  the  slope,  as 
well  as  with  the  roughness  of  the  surface.  To  determine 
the  value  of  c  for  a  particular  case  the  quantities  v,  r,  and  s 
are  measured,  and  then  c  is  computed.  To  find  r  and  s 
linear  measurements  and  leveling  are  required.  To  deter- 
mine v  the  flow  must  be  gaged  either  in  a  measuring  vessel 
or  by  an  orifice  or  weir,  or,  if  the  channel  be  large,  by  floats 
or  other  indirect  methods  described  in  the  next  chapter, 
and  then  the  mean  velocity  v  is  computed  from  v=q/a. 
It  being  a  matter  of  great  importance  to  establish  a  satis- 
factory formula  for  mean  velocity,  thousands  of  such 
gagings  have  been  made,  and  from  the  records  of  these 
the  values  of  the  coefficients  given  in  the  tables  at  the  end 
of  this  volume  and  in  the  following  articles  have  been 
deduced. 

Prob.  106.  Compute  the  value  of  c  for  a  circular  masonry 
conduit  4  feet  in  diameter  which  delivers  29  cubic  feet  per 
second  when  running  half  full,  its  slope  or  grade  being  1.5  feet 
in  1000  feet. 

ART.  107.     CIRCULAR  CONDUITS,  FULL  OR  HALF  FULL 

When  a  circular  conduit  of  diameter  d.  runs  either  full 
or  half  full  of  water  the  hydraulic  radius  is  %d,  and  the 
Chezy  formula  for  mean  velocity  is 

v  =  cVrs  =  c  .  -JVS 

The  velocity  can  then  be  computed  when  c  is  known,  and 
for  this  purpose  Table  37  gives  Hamilton  Smith's  values 
of  c  for  pipes  and  conduits  having  quite  smooth  interior 
surfaces,  and  no  sharp  bends.*  The  discharge  per  second 
then  is 


in  which  a  is  either  the  area  of  the  circular  cross-section  or 
one-half  that  section,  as  the  case  may  be. 

"*"  ^vdraulics  (London  and  New  York,  1886),  p.  271. 


ART.  107     CIRCULAR  CONDUITS,  FULL  OR  HALF  FULL         271 

To  use  Table  37  a  tentative  method  must  be  employed, 
since  c  depends  upon  the  velocity  of  flow.  For  this  purpose 
there  may  be  taken  roughly 

mean  Chezy  coefficient  0  =  125 

and  then  v  may  be  computed  for  the  given  diameter  and 
slope  ;  a  new  value  of  c  is  then  taken  from  the  table  and  a 
new  v  computed;  and  thus,  after  two  or  three  trials,  the 
probable  mean  velocity  of  flow  is  obtained.  The  value  of 
d  must  be  expressed  in  feet. 

For  example,  let  it  be  required  to  find  the  velocity  and 
discharge  of  a  semicircular  conduit  of  6  feet  diameter  when 
laid  on  a  grade  of  o.i  feet  in  100  feet.  First, 


v  =  12  $X%V  6X0.001  =4.8  feet  per  second. 
For  this  velocity  the  table  gives  147  for  c  ;  hence 

T;  =  147  XjV  '0.006  =  5.7  feet  per  second. 
Again,  from  the  table  c  =  150,  and 

v  =  150  X  jVo.oo6  =  5-8  feet  per  second. 

This  shows  that  150  is  a  little  too  large;  for  c  =  149.5,  *;  is 
found  to  be  5.79  feet  per  second,  which  is  the  final  result. 
The  discharge  per  second  now  is 

q  =  o.  7854X4X36X5.  79  =81.9  cubic  feet 
which  is  the  probable  flow  under  the  given  conditions. 

To  find  the  diameter  of  a  circular  conduit  to  discharge  a 
given  quantity  under  a  given  slope,  the  area  a  is  to  be  ex- 
pressed in  terms  of  d  in  the  above  equation,  which  is  then 
to  be  solved  for  d\  thus, 


the  first  being  for  a  conduit  running  full  and  the  second  for 
one  running  half  full.  Here  c  may  at  first  be  taken  as  125  ; 
then  d  is  computed,  the  approximate  velocity  found  from 
2  and  with  this  value  of  v  a  value  of  c  is  selected 


272  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

from  the  table,  and  the  computation  for  d  is  repeated.  This 
process  may  be  continued  until  the  corresponding  values 
of  c  and  v  are  found  to  be  in  close  agreement. 

As  an  example  of  the  determination  of  diameter  let  it 
be  required  to  find  d  when  9  =  81.9  cubic  feet  per  second, 
^=0.001,  and  the  conduit  runs  full.  For  0  =  125  the  f°r- 
mula  gives  ^  =  4.9  feet,  whence  ^  =  4.37  feet  per  second. 
From  the  table  c  may  be  now  taken  as  142,  and  repeating 
the  computation  ^  =  4.64  feet,  whence  ^=4.84  feet  per 
second,  which  requires  no  further  change  in  the  value  of  c. 
As  the  tabular  coefficients  are  based  upon  quite  smooth 
interior  surfaces,  such  as  occur  only  in  new,  clean  iron  pipes, 
or  with  fine  cement  finish,  it  might  be  well  to  build  the 
conduit  5  feet  or  60  inches  in  diameter.  It  is  seen  from 
the  previous  example  that  a  semicircular  conduit  of  6  feet 
diameter  carries  the  same  amount  of  water  as  is  here  pro- 
vided for. 

Circular  conduits  running  full  of  water  are  long  pipes  and 
all  the  formulas  and  methods  of  Arts.  90  and  91  can  be 
applied  also  to  their  discussion.  From  Art.  106  it  is  seen 
that 

c=\/8g77         or         c  =  i6.04/v/f 

in  which  /  is  to  be  taken  from  Table  33.  Values  of  c  com- 
puted in  this  manner  will  not  generally  agree  closely  with 
the  coefficients  of  Smith,  partly  because  the  values  of  /  are 
given  only  to  three  decimal  places,  and  partly  because 
Table  33  for  pipes  was  constructed  from  experiments  on 
smoother  surfaces  than  those  of  conduits.  An  agreement 
within  5  percent  in  mean  velocities  deduced  by  different 
methods  is  all  that  can  generally  be  expected  in  conduit  com- 
putations, and  if  the  actual  discharge  agrees  as  closely  as 
this  with  the  computed  discharge,  the  designer  can  be  con- 
sidered a  fortunate  man. 

All  of  the  laws  deduced  in  the  last  chapter  regarding  the 
relation  between  diameter  and  discharge,  relative  discharg- 


ART.  ios  CIRCULAR  CONDUITS,  PARTLY  FULL  273 

ing  capacity,  etc.,  hence  apply  equally  well  to  circular  con- 
duits which  run  either  full  or  half  full.  If  the  conduit  be 
full  it  matters  not  whether  it  be  laid  truly  to  grade  or  whether 
it  be  under  pressure,  since  in  either  "case  the  slope  s  is  the 
total  fall  h  divided  by  the  total  length.  Usually,  however, 
the  word  conduit  implies  a  uniform  slope  for  considerable 
distances,  and  in  this  case  the  hydraulic  gradient  coincides 
with  the  surface  of  the  flowing  water. 

Prob.  107a.  Find  the  discharge  of  a  conduit  when  running 
full,  its  diameter  being  6  feet  and  its  fall  9.54  feet  in  one  mile. 

Prob.  1076.  Find  the  diameter  of  a  conduit  to  deliver  when 
running  full  16  500  ooo  gallons  per  day,  its  slope  being  0.00016. 

ART.  108.     CIRCULAR  CONDUITS,  PARTLY  FULL 

Let  a  circular  conduit  with  the  slope  5  be  partly  full 
of  water,  its  cross-section  being  a  and  hydraulic  radius 
r.  Then  the  mean  velocity  and  the  discharge  are  given  by 

v  =  c\/rs  q  =  ca\/rs 

The  mean  velocity  is  hence  proportional  to  V 'r  and  the 
discharge  to  aVr,  provided  that  c  be  a  constant.  Since, 
however,  c  varies  slightly  with  r,  this  law  of  proportion- 
ality is  not  exact  but  approximate. 

When  a  circular  conduit  of  diameter  d  runs  either  full 
or  half  full  its  hydraulic  radius  is  \d  (Art.  105).  If  it  is 
filled  to  the  depth  d' ',  the  wetted  perim- 
eter is 

.   2d'-d 
p  =  \iid  +  a  arc  sin — -7 — 

and    the    sectional    area    of    the    water 

surface  is  

a=±dp  +  (d'  -  ±-d)  Vdf  (d  -  d') 

From  these  p  and  a  can  be  computed,  and  then  r  is  found 
by  dividing  a  by  p.  Table  39  gives  values  of  p,  a,  and  r  for 


274  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

a  circle  of  diameter  unity  for  different  depths  of  water. 
To  find  from  it  the  hydraulic  radius  for  any  other  circle  it 
is  only  necessary  to  multiply  the  tabular  values  of  r  by  the 
given  diameter  d.  The  table  shows  that  the  greatest 
value  of  the  hydraulic  radius  occurs  when  d'=o.8id, 
and  that  it  is  but  little  less  when  d' =o.8d.  In  the  fifth 
and  sixth  columns  of  the  table  are  given  values  of  VV 
and  aVr  for  different  depths  in  the  circle  of  diameter 
unity ;  these  are  approximately  proportional  to  the  veloc- 
ity and  discharge  which  occur  in  a  circle  of  any  size. 
The  table  shows  that  the  greatest  velocity  occurs  when 
the  depth  of  the  water  is  about  eight-tenths  of  the  di- 
ameter, and  that  the  greatest  discharge  occurs  when  the 
depth  is  about  0.95^,  or  £-§-  of  the  diameter. 

By  the  help  of  Table  39  the  velocity  and  discharge 
may  be  computed  when  c  is  known,  but  it  is  not  possible 
on  account  of  the  lack  of  experimental  knowledge  to 
state  precise  values  of  c  for  different  values  of  r  in  circles 
of  different  sizes.  However,  it  is  known  that  an  increase 
in  r  increases  c,  and  that  a  decrease  in  r  decreases  c.  The 
following  experiments  of  Darcy  and  Bazin  show  the  extent 
of  this  variation  for  a  semicircular  conduit  of  4.1  feet 
diameter,  and  they  also  teach  that  the  nature  of  the  in- 
terior surface  greatly  influences  the  values  of  c.  Two 
conduits  were  built;  each  with  a  slope  s  =0.0015  an<^  d  =  4.i 
feet.  One  was  lined  with  neat  cement,  and  the  other 
with  a  mortar  made  of  cement  with  one-third  fine  sand. 
The  flow  was  allowed  to  occur  with  different  depths, 
and  the  discharges  per  second  were  gaged  by  means  of 
orifices;  this  enabled  the  velocities  to  be  computed,  and 
from  these  the  values  of  the  coefficient  c  were  found.  The 
following  are  a  portion  of  the  results  obtained,  df  denoting 
the  depth  of  water  in  the  conduit,  r  the  hydraulic  radius, 
v  the  mean  velocity,  and  all  linear  dimensions  being  in. 
English  feet: 


ART.  109  RECTANGULAR  CONDUITS  275 

For  cement  lining.  For  mortar  lining. 

df              r                v  C                         d'              r                v  C 

2.05        1.029       6.06  154  2.04        1.022        5.55  142 

1.61       0.867       5-29  147  1-69       0.900       4.94  135 

1.03       0.605       4-i6  138  1.09       0.635       3-87  125 

°-59       0.366       3.02  129  0.61       0.379       2.87  120 

It  is  here  seen  that  c  decreases  quite  uniformly  with  r, 
and  that  the  velocities  for  the  mortar  lining  are  8  or  10 
percent  less  than  those  for  the  neat  cement  lining. 

The  value  of  the  coefficient   c  for  these  experiments 
may  be  roughly  expressed  for  English  measures  by 


in  which  ct  is  the  coefficient  for  the  conduit  when  running 
half  full.  How  this  will  apply  to  different  diameters 
and  velocities  is  not  known;  when  d'  is  greater  than  o.&d 
it  will  probably  prove  incorrect.  In  practice,  however, 
computations  on  the  flow  in  partly  filled  conduits  are  of 
rare  occurrence. 

Prob.  108.  Compute  the  hydraulic  radius  for  a  circular  con- 
duit when  it  is  three-fourths  filled  with  water,  and  also  the  mean 
velocity  if  it  be  lined  with  neat  cement  and  laid  on  a  grade  of 
0.15  per  100,  the  diameter  being  4.1  feet. 

ART.  109.     RECTANGULAR  CONDUITS 

In  designing  an  open  rectangular  trough  or  conduit 
to  carry  water  there  is  a  certain  ratio  of  breadth  to  depth 
which  is  most  advantageous,  because  thereby  either  the 
discharge  is  the  greatest  or  the  least  amount  of  material 
is  required  for  its  construction.  Let  b  be  the  breadth 
and  d  the  depth  of  the  water  section,  then  the  area  a  is 
bd  and  the  wetted  perimeter  p  is  b  +  2d.  If  the  area  a  is 
given  it  may  be  required  to  find  the  relation  between  b 
and  d  so  that  the  discharge  may  be  a  maximum.  If  the 
wetted  perimeter  p  is  given,  the  relation  between  b  and 
d  to  produce  the  same  result  may  be  demanded.  It  is 
now  to  be  shown  that  in  both  cases  the  breadth  is  double 


276  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

the  depth,  or  b  =  2d.  This  is  called  the  most  advantageous 
proportion  for  an  open  rectangular  conduit,  since  there 
is  the  least  head  lost  in-  friction  when  the  velocity  and 
discharge  are  the  greatest  possible. 

Let  r  be  the  hydraulic  radius  of  the  cross-section,  or 

a       bd 

~P=b+2d 

then,  from,  the  Chezy  formula  (106),  the  expressions  for 
the  velocity  and  discharge  are 


bd 


In  these   expressions  it   is  required  to   find  the  relation 
between  b  and  d,  which  renders  both  v  and  q  a  maximum. 

Let  the  wetted  perimeter  p  be  given,  as  might  be  the 
case  when  a  definite  amount  of  lumber  is  assigned  for  the 
construction  of  a  trough;  then  b  +  2d=p,  or  d  =  %(p  —  b), 
and 

Ib(p-b)  /-   \b\p-b 


in  which  p  is  a  constant.  Differentiating  either  of  these 
expressions  with  respect  to  b  and  equating  the  derivative 
to  zero,  there  is  found  b  =  \p,  and  hence  d  =  \p.  Accord- 
ingly b  =  2d,  or  the  breadth  is  double  the  depth. 

Again,  let  the  area  a  be  given,  as  might  be  the  case 
when  a  definite  amount  of  rock  excavation  is  to  be  made  ; 
then  bd=a,  or  d  =  a/b,  and 


-    I    asb 


in  which  a  is  constant.  By  equating  the  first  derivative 
to  zero  there  is  found  62  =  2a,  and  hence  d2=%a.  Accord- 
ingly b  =  2d,  or  the  breadth  is  double  tho  depth,  as  before. 

It  is  seen  in  the   above   cases  that  the  maximum  of 
both  v  and  q  occur  when  r  is  a  maximum,  or  when  r  =  J d. 


ART.  109  RECTANGULAR  CONDUITS  277 

It  is  indeed  a  general  rule  that  r  should  be  a  maximum 
in  order  to  secure  the  least  loss  of  head  in  friction.  The 
circle  has  a  greater  hydraulic  radius  than  any  other  figure 
of  equal  area. 

In  these  investigations  c  has  been  regarded  as  constant, 
although  strictly  it  varies  somewhat  for  different  ratios 
of  b  to  d.  The  rule  deduced  is,  however,  sufficiently 
close  for  all  practical  purposes.  It  frequently  happens 
that  it  is  not  desirable  to  adopt  the  relation  b  =  2 d,  either 
because  the  water  pressure  on  the  sides  of  the  conduit 
becomes  too  great  or  because  it  is  desirable  to  limit  the 
velocity  so  as  to  avoid  scouring  the  bed  of  the  channel. 
Whenever  these  considerations  are  more  important  than 
that  of  securing  the  greatest  discharge  the  depth  is  made 
less  than  one-half  the  breadth. 

The  velocity  and  discharge  through  a  rectangular  con- 
duit are  expressed  by  the  general  equations 

v  =  c\/rs  q=av  =  ca\/rs 

and  are  computed  without  difficulty  for  any  given  case 
when  the  coefficient  c  is  known.  To  ascertain  this,  how- 
ever, is  not  easy,  for  it  is  only  from  recorded  experiments 
that  its  value  can  be  ascertained.  When  the  depth  of 
the  water  in  the  conduit  is  one-half  of  its  width,  thus 
giving  the  most  advantageous  section,  the  values  of  c 
for  smooth  interior  surfaces  may  be  estimated  by  the 
use  of  Table  37  for  circular  conduits,  although  c  is  prob- 
ably smaller  for  rectangles  than  for  circles  of  equal  area. 
When  the  depth  of  the  water  is  less  or  greater  than  \d, 
it  must  be  remembered  that  c  increases  with  r.  The 
value  of  c  also  is  subject  to  slight  variations  with  the 
slope  5,  and  to  great  variations  with  the  degree  of  rough- 
ness of  the  surface. 

Table  40,  derived  from  Smith's  discussion  of  the 
experiments  of  Darcy  and  Bazin,  gives  values  of  c  for 


278  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

a  number  of  wooden  and  masonry  conduits  of  rectangular 
sections,  all  of  which  were  laid  on  the  grade  of  0.49  percent 
or  5=0.0049.  The  great  influence  of  roughness  of  surface 
in  diminishing  the  coefficient  is  here  plainly  seen.  For 
masonry  conduits  with  hammer-dressed  surfaces  c  may  be 
as  low  as  60  or  50,  particularly  when  covered  with  moss 
and  slime. 

Prob.  109a.  Compare  the  discharge  of  a  trough  3  feet  wide 
and  i  foot  deep  with  that  of  two  troughs  each  1.5  feet  wide  and 
i  foot  deep. 

Prob.  1096.  Find  the  size  of  a  trough,  whose  width  is  double 
its  depth,  which  will  deliver  125  cubic  feet  per  minute  when  its 
slope  is  0.002,  taking  c  as  100. 

ART.  110.     TRAPEZOIDAL  SECTIONS 

Ditches  and  conduits  are  often  built  with  a  bottom 
nearly  flat  and  with  side  slopes,  thus  forming  a  trapezoidal 
section.  The  side  slope  is  fixed  by  the  nature  of  the  soil 
or  by  other  circumstances,  the  grade  is  given,  and  it  may 
be  then  required  to  ascertain  the  relation  between  the  bot- 
tom width  and  the  depth  of  water,  in  order  that  the  section 
shall  be  the  most  advantageous.  This  can  be  done  by  the 
same  reasoning  as  used  for  the  rectangle  in  the  last  article, 
but  it  may  be  well  to  employ  a  different  method,  and  thus 
be  able  to  consider  the  subject  in  a  new  light. 

Let  the  trapezoidal  channel  have  the  bottom  width  b, 
the  depth  d,  and  let  6  be  the  angle  made  by  the  side  slopes 

with  the  horizontal.  Let  it  be 
required  to  discharge  q  cubic 
units  of  water  per  second.  Now 
q  =  caVrs  and  the  most  advan- 
FIG  11Q  tageous  proportions  may  be  said 

to  be  those  that  will  render  the 

cross-section  a  a  minimum  for  a  given  discharge,  for  thus 
the  least  excavation  will  be  required.  From  the  figure 

)         p  =  b  +  2  d/sind 


ART.  no  TRAPEZOIDAL  SECTIONS  279 

and  from  these  the  value  of  r  may  be  expressed  in  terms  of 
a,  d,  and  6  ;  inserting  this  in  the  formula  for  q,  it  reduces  to 


C2sa3     q*a       J 
-  --       = 


l  Q 
\sin0 

in  which  the  second  member  is  a  constant.  Obtaining  the 
first  derivative  of  a  with  respect  to  d,  and  then  replacing 
q2  by  its  value  cza*rs,  there  results 


which  is  the  relation  that  renders  the  area  a  a  minimum, 
that  is,  the  advantageous  depth  is  double  the  hydraulic 
radius.  Now  since  a/p  =r  it  is  easy  to  show  that 


or,  the  top  width  of  the  water  surface  should  equal  the 
sum  of  the  two  side  slopes  in  order  to  give  the  most  advan- 
tageous section.  Since  c  has  been  regarded  constant  the 
conclusion  is  not  a  rigorous  one,  although  it  may  safely  be 
followed  in  practice.  As  in  all  cases  of  an  algebraic  mini- 
mum, a  considerable  variation  in  the  value  of  the  ratio  d/b 
may  occur  without  materially  affecting  the  value  of  the  area 
a.  In  many  cases  it  is  not  possible  to  have  so  great  a  depth 
of  water  as  the  rule  d  =  2r  requires  because  of  the  greater 
cost  of  excavation  at  such  depth,  or  because  width  rather 
than  depth  may  be  needed  for  other  reasons. 

When  a  trapezoidal  channel  is  to  be  built  the  general 
formulas  v  =cv/rs  and  q=av  may  be  used  to  obtain  a  rough' 
approximation  to  the  discharge,  c  being  assumed  from  the 
best  knowledge  at  hand.'  The  formula  of  Kutter  (Art.  Ill) 
may  be  used  to  determine  c  when  the  nature  of  the  bed 
of  the  channel  is  known.  For  a  channel  already  built,  com- 
putations cannot  be  trusted  to  give  reliable  values  of  the 
.discharge  on  account  of  the  uncertainty  regarding  the 
coefficient,  and  in  an  important  case  an  actual  gaging  of 
the  flow  should  be  made.  This  is  best  effected  by  a  weir, 


280  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

but  if  that  should  prove  too  expensive,  the  methods  ex- 
plained in  the  next  chapter  may  be  employed  to  give  more 
precise  results  than  can  usually  be  determined  by  computa- 
tion from  any  formula. 

The  problem  of  determining  the  size  of  a  trapezoidal 
channel  to  carry  a  given  quantity  of  water,  does  not  require 
c  to  be  determined  with  great  precision,  as  an  allowance 
should  be  made  on  the  side  of  safety.  For  this  purpose 
the  following  values  may  be  used,  the  lower  ones  being  for 
small  cross-sections  with  rough  and  foul  surfaces,  and  the 
higher  ones  for  large  cross-sections  with  quite  smooth  and 
clean  earth  surfaces: 

For  unplaned  plank,  c  =  100  to  120 

For  smooth  masonry,  c  =    90  to  1  10 

For  clean  earth,  c  =    60  to    80 

For  stony  earth,  c  =   40  to    60 

For  rough  stone,  c=   35  to    50 

For  earth  foul  with  weeds,  c  =   30  to    50 

To  solve  this  problem,  let  a  and  p  be  replaced  by  their  values 
in  terms  of  b  and  d.  The  discharge  then  is 


Now  when  q,  c,  0,  and  5  are  known,  the  equation  contains 
two  unknown  quantities,  b  and  d.  If  the  section  is  to  be 
the  most  advantageous,  b  can  be  replaced  by  its  value  in 
terms  of  d  as  above  found,  and  the  equation  then  has  but 
one  unknown.  Or  in  general,  if  b=md,  where  m  is  any 
assumed  number,  a  solution  for  the  depth  gives  the  formula 

_      q\msmd  +  2) 

~c2s(m  +  cotd)3  sin  '6 

For  the  particular  case  where  the  side  slopes  are  i  on  i  or 
0  =  45°,  and  the  bottom  width  is  to  be  equal  to  the  water 
depth,  or  m  =  i  ,'  this  becomes 

4  =  0.863  (22/c2<r)* 


ART.  ill  KUTTER'S  FORMULA  281 

These  formulas  are  analogous  to  those  for  finding  the  diam- 
eter of  pipes  and  circular  conduits,  and  the  numerical  oper- 
ations are  in  all  respects  similar.  It  is  plain  that  by  assign- 
ing different  values  to  m  numerous  sections  may  be  deter- 
mined which  will  satisfy  the  imposed  conditions,  and  usually 
the  one  is  to  be  selected  that  will  give  both  a  safe  velocity 
and  a  minimum  cost.  In  Art.  113  will  be  found  an  example 
of  the  determination  of  the  size  of  a  trapezoidal  canal. 

Prob.  llOa.  For  the  most  advantageous  trapezoidal  cross- 
section  show  that  the  area  is  d2(2  —  cos#)/sin#,  and  that  the  bot- 
tom width  is  2d  tan W.. 

Prob.  1106.  If  the  value  of  c  is  71,  compute  the  depth  of  a 
trapezoidal  section  to  carry  200  cubic  feet  of  water  per  second, 
0  being  45°,  the  slope  5  being  o.ooi  and  the  bottom  width  being 
equal  to  the  depth.  Compute  also  the  area  of  the  cross-section 
and  the  mean  velocity. 

ART.  111.     KUTTER'S  FORMULA 

An  elaborate  discussion  of  all  recorded  gagings  of  chan- 
nels was  made  by  Ganguillet  and  Kutter  in  1869,  from 
which  an  important  empirical  formula  was  deduced  for  the 
coefficient  c  in  the  Chezy  formula  v=cVrs.  The  value  of 
c  is  expressed  in  terms  of  the  hydraulic  radius/,  the  slope  s, 
and  the  degree  of  roughness  of  the  surface,  and  may  be 
computed  when  these  three  quantities  are  given.  When 
r  is  in  feet  and  v  in  feet  per  second,  Kutter 's  formula  is 

1.811  0.00281 

——  +  41.65+-  — 

(HI) 


n 


in  which  n  is  an  abstract  number  whose  value  depends  only 
upon  the  roughness  of  the  surface.  v  By  inserting  this  value 
of  c  in  the.  Chezy  formula  for  v,  the  mea«i  velocity  is  made 
to  depend  upon  r,  s,  and  the  roughness  of  the  surface.  The 


282  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

following  are  the  values  of  n  assigned  by  Kutter  to  different 
surfaces : 

n  =  0.009  for  well-planed  timber, 

n  =  o.oio  for  neat  cement, 

n  =  o.on  for  cement  with  one-third  sand, 

n  =  0.012  for  unplaned  timber, 

n  =  0.013  f°r  ashlar  and  brick  work, 

n  =  0.015  for  unclean  surfaces  in  sewers  and  conduits, 

^  =  0.017  for  rubble  masonry, 

n  —  0.020  for  canals  in  very  firm  gravel, 

^  =  0.025  for  canals  and  rivers  free  from  stones  and  weeds, 

n  —  0.030  for  canals  and  rivers  with  some  stones  and  weeds, 

^  =  0.035  for  canals  and  rivers  in  bad  order. 

The  formula  of  Kutter  has  received  a  wide  acceptance 
on  account  of  its  application  to  all  kinds  of  surfaces.  Not- 
withstanding that  it  is  purely  empirical,  and  hence  not 
perfect,  it  is  to  be  regarded  as  a  formula  of  great  value,  so 
that  no  design  for  a  conduit  or  channel  should  be  completed 
without  employing  it  in  the  investigation,  even  if  the  final 
construction  be  not  based  upon  it.  In  sewer  work  it  is 
extensively  employed,  n  being  taken  as  about  0.015.  The 
formula  shows  that  the  coefficient  c  always  increases  with 
r,  that  it  decreases  with  5  when  r  is  greater  than  3.28  feet, 
and  that  it  increases  with  5  when  r  is  less  than  3.28  feet. 
When  r  equals  3.28  feet  the  value  of  c  is  simply  i.Sn/n. 
It  is  not  likely  that  future  investigations  will  confirm  these 
laws  of  variation  in  all  respects. 

In  the  following  articles  are  given  values  of  c  for  a  few 
cases,  and  these  might  be  greatly  extended,  as  has  been 
done  by  Kutter  and  others.*  But  this  is  scarcely  necessary 
except  for  special  lines  of  investigation,  since  for  single  cases 
there  is  no  difficulty  in  directly  computing  it  for  given  data. 
For  instance,  take  a  rectangular  trough  of  unplaned  plank 
3.93  feet  wide  on  a  slope  of  4.9  feet  in  1000  feet,  the  water 

*  Flow  of   Water   in   Rivers   and   Other   Channels.      Translated,  with 
additions,  by  Hering  and  Trautwine,  New  York,  1889. 


ART.  112  SEWERS  283 

being  1.29  feet  deep.     Here  5=0.0049,  and  r  =  0.779  feet. 
Then  n  being  0.012,  the  value  of  c  is  found  to  be 

1.811  0.00281 

-  +  41.65+- 
0.012  J      0.0049 

/~\    f~\  T  r\         I  /^    /"\  x%  /**  Q  T  \  *J 


1  + 


Vo.779 


/  0.0028l\ 

41.65  +  - 
\*         0.0049  / 


The  data  here  used  are  taken  from  Table  40,  where  the  actual 
value  of  c  is  given  as  117 ;  hence  in  this  case  Kutter's  for- 
mula is  about  5  percent  in  excess.  As  a  second  example,  the 
following  data  from  the  same^  table  will  be  taken :  a  rect- 
angular conduit  in  neat  cement,  6  =  5.94  feet,  ^  =  0.91  feet, 
5=0.0049.  Here  #=o.oio,  and  r=o.6q'j  feet.  Inserting 
all  values  in  the  formula,  there  is  found  c  =  148,  which  is  8 
percent  greater  than  the  true  value,  138.  Thus  is  shown 
the  fact  that  errors  of  5  and  10  percent  are  to  be  regarded 
as  common  in  calculations  on  the  flow  of  water  in  conduits 
and  canals. 

Prob.  111.  The  Sudbury  conduit  is  of  horse-shoe  form  and 
lined  with  brick  laid  with  cement  joints  one-quarter  of  an  inch 
thick,  and  laid  on  a  slope  of  0.0001895.  Compute  the  discharge 
in  24  hours  when  the  area  is  33.31  square  feet  and  the  wetted 
perimeter  15.21  feet. 

ART.  112.     SEWERS  ' 

Sewers  smaller  in  diameter  than  18  inches  are  always 
circular  in  section.  When  larger  than  this  they  are  built 
with  the  section  either  circular,  egg-shaped,  or  of  the  horse- 
shoe form.  The  last  shape  is  very  disadvantageous  when  a 
small  quantity  of  sewage  is  flowing,  for  the  wetted  perimeter 
is  then  large  compared  with  the  area,  the  hydraulic  radius 
is  small,  and  the  velocity  becomes  low,  so  that  a  deposit  of 
the  foul  materials  results.  As  the  slope  of  sewer  lines  is 
often  very  slight,  it  is  important  that  such  a  form  of  cross- 
section  should  be  adopted  to  render  the  velocity  of  flow 


284 


FLOW  IN  CONDUITS  AND  CANALS 


CHAP.  IX 


sufficient  to  prevent  this  deposit.  A  velocity  of  2  feet  per 
second  is  found  to  be  about  the  minimum  allowable  limit, 
and  4  feet  per  second  need  not  be  usually  exceeded. 

The  egg-shaped  section  is  designed  so  that  the  hydraulic 
radius  may  not  become  small  even  when  a  small  amount  of 

sewage  is  flowing.  One  of  the 
most  common  forms  is  that 
shown  in  Fig.  112,  where  the 
greatest  width  DD  is  two-thirds 
of  the  depth  HM.  The  arch 
DHD  is  a  semicircle  described 
from  A  as  a  center.  The  invert 
LML  is  a  portion  of  a  circle  de- 
scribed from  B  as  a  center,  the 
distance  B  A  being  three-fourths 
of  DD  and  the  radius  BM  being  one-half  of  AD.  Each  side 
DL  is  described  from  a  center  C  so  as  to  be  tangent  to  the 
arch  and  invert.  These  relations  may  be  expressed  more 
concisely  by 

HM  =  i^D       AB=ID        BM  =  \D        CL  =  i±D 
in  which  D  is  the  horizontal  diameter  DD\ 

Computations  on  egg-shaped  sewers  are  usually  confined 
to  three  cases,  namely,  when  flowing  full,  two-thirds  full, 
and  one-third  full.  The  values  of  the  sectional  areas, 
wetted  perimeters,  and  hydraulic  radii  for  these  cases,  as 
given  by  Flynn,*  are 

apt 
Full  1. 1485!)'         3.965!) 

Two-thirds  full     o.^^D2         2.394!} 
One-third  full       o. 2840^2          1.375!) 


0.2897!) 
0.3157!) 
0.2066!) 


This  shows  that  the  hydraulic  radius,  and  hence  the  velocity, 
is  but  little  less  when  flowing  one-third  full  than  when 
flowing  with  full  section. 

*  Van  Nostrand's  Magazine,  1883,  vol.  28,  p.  138. 


ART.  112  SEWERS  285 

Egg-shaped  sewers  and  small  circular  ones  are  formed  by 
laying  consecutive  lengths  of  clay  or  cement  pipe  whose  in- 
terior surfaces  are  quite  smooth  when  new,  but  may  become 
foul  after  use.  Large  sewers  of  circular  section  are  made 
of  brick,  and  are  more  apt  to  become  foul  than  smaller  ones. 
In  the  separate  system,  where  systematic  flushing  is  em- 
ployed and  the  pipes  are  small,  foulness  of  surface  is  not 
so  common  as  in  the  combined  system,  where  the  storm 
water  is  alone  used  for  this  purpose.  In  the  latter  case 
the  sizes  are  computed  for  the  volume  of  storm  water  to  be 
discharged,  the  amount  of  sewage  being  very  small  in  com- 
parison. 

The  discharge  of  a  sewer  pipe  enters  it  at  intervals  along 
its  length,  and  hence  the  flow  is  not  uniform.  .  The  depth  of 
the  flow  increases  along  the  length,  and  at  junctions  the  size 
of  the  pipe  is  enlarged.  The  strict  investigation  of  the 
problem  of  flow  is  accordingly  one  of  great  complexity. 
But  considering  the  fact  that  the  sewer  is  rarely  filled,  and 
that  it  should  be  made  large  enough  to  provide  for  contin- 
gencies and  future  extensions,  it  appears  that  great  pre- 
cision is  unnecessary.  The  practice,  therefore,  is  to  discuss 
a  sewer  for  the  condition  of  maximum  discharge,  regarding 
it  as  a  channel  with  uniform  flow.  The  main  problem  is 
that  of  the  determination  of  size;  if  the  form  be  circular, 
the  diameter  is  found,  as  in  Art.  107,  by 

d  =  (8g/;rc  V7)*  =  1.45  (g/c\/f)* 

If  the  form  be  egg-shaped  and  of  the  proportions  above  ex- 
plained, the  discharge  when  running  full  is 

g  =  ac  Vrs  ==  i  .  i485L>2c  Vo.  2897!).? 
from  which  the  value  of  D  is  found  to  be 


Thus,  when  q  has  been  determined  and  c  is  known,  the  re- 
quired sizes  for  given  slopes  can  be  computed.     The  velocity 


286  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

should  also  be  found  in  order  to  ascertain  if  it  be  low  enough 
to  prevent  scouring  (Art.  127). 

Experiments  from  which  to  directly  determine  the  co- 
efficient c  for  the  flow  in  sewers  are  few  in  number,  but 
since  the  sewage  is  mostly  water,  it  may  be  approximately 
ascertained  from  the  values  for  similar  surfaces.  Kutter's 
formula  has  been  extensively  employed  for  this  purpose, 
using  0.015  for  the  coefficient  of  roughness.  Table  42  gives 
values  of  c  for  three  different  slopes  and  for  two  classes  of 
surfaces.  The  values  for  the  degree  of  roughness  repre- 
sented by  ^=0.017  are  applicable  to  sewers  with  quite 
rough  surfaces  of  masonry;  those  for  7^  =  0.015  are  appli- 
cable to  sewers  with  ordinary  smooth  surfaces,  somewhat 
fouled  or  tuberculated  by  deposits,  and  are  the  ones  to  be 
generally  used  in  computations.  By  the  help  of  this  table 
and  the  general  equations  for  mean  velocity  and  discharge, 
all  problems  relating  to  flow  in  sewers  can  be  readily  solved. 

Prob.  112.  The  grade  of  a  sewer  is  one  foot  in  960,  and  its 
discharge  is  to  be  65  cubic  feet  per  second.  What  is  the  diam- 
eter of  the  sewer  if  circular? 

ART.  113.     DITCHES  AND  CANALS 

Ditches  for  irrigating  purposes  are  of  a  trapezoidal  sec- 
tion, and  the  slope  is  determined  by  the  fall  between  the 
point  from  which  the  water  is  taken  and  the  place  of  de- 
livery. If  the  fall  is  large  it  may  not  be  possible  to  con- 
struct the  ditch  in  a  straight  line  between  the  two  points,, 
even  if  the  topography  of  the  country  should  permit,  on 
account  of  the  high  velocity  which  would  result.  A  veloc- 
ity exceeding  2  feet  per  second  may  often  injure  the  bed  of 
the  channel  by  scouring,  unless  it  be  protected  by  riprap 
or  other  lining.  For  this  reason,  as  well  as  for  others,  the 
alignment  of  ditches  and  canals  is  often  circuitous. 

The  principles  of  the  preceding  articles  are  sufficient  to 
solve  all  usual  problems  of  uniform  flow  in  such  channels 


ART.  113  DITCHES  AND  CANALS  287 

when  the  values  of  the  Chezy  coefficient  c  are  known.  These 
are  perhaps  best  determined  by  Kutter's  formula,  and  for 
greater  convenience  Table  44  has  been  prepared  which  gives 
their  values  for  three  slopes  and  two  degrees  of  roughness. 
By  interpolation  in  this  table  values  for  intermediate  data 
may  also  be  found;  for  instance,  if  the  hydraulic  radius 
be  3.5  feet,  the  slope  be  i  on  1000,  and  n  be  0.025,  the 
value  of  c  is  found  to  be  74.5. 

As  an  example  of  the  use  of  the  table  let  it  be  required 
to  find  the  width  and  depth  of  a  ditch  of  most  advantageous 
cross-section,  whose  channel  is  to  be  in  tolerably  good  order, 
so  that  ^=0.025.  The  amount  of  water  to  be  delivered  is 
200  cubic  feet  per  second  and  the  grade  is  i  in  1000,  the 
side  slopes  of  the  channel  being  i  on  i.  From  Art.  110  the 
relation  between  the  bottom  width  and  the  depth  of  the 
water  is,  since  the  angle  6  is  45°, 

/     2  \ 

b  =d(  -r—jr-2  cot  6    =0.828^ 
\sm  6  } 

The  area  of  the  cross-section  then  is 

and  the  wetted  perimeter  of  the  cross-section  is 


whence  the  hydraulic  radius  is  o.$d,  as  must  be  the  case 
for  all  trapezoidal  channels  of  most  advantageous  section. 
Now,  since  d  is  unknown,  c  cannot  be  taken  from  the  table, 
and  as  a  first  approximation  let  it  be  supposed  to  be  60. 
Then  in  the  general  formula  for  discharge  the  above  values 
are  substituted,  giving 

200  =6oX  i.  828d2>/o.  5^X0.001 

from  which  d  is  found  to  be  5.8  feet.  Accordingly  ^  =  2.9 
feet,  and  from  the  table  c  is  about  71.  Repeating  the  com- 


288  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

putation  with  this  value  of  c  there  is  found  ^  =  5.44  feet, 
which,  considering  the  uncertainty  of  c,  is  sufficiently  close. 
The  depth  may  then  be  made  5.5  feet,  the  bottom  width  is 

6=0.828X5.5  =4-55  feet 
and  the  area  of  the  cross-section  is 

a  =  i.828X5.52  =  55-3  square  feet 

+ 
which  gives  for  the  mean  velocity 

v= •  =3.62  feet  per  second 

This  completes  the  investigation  if  the  velocity  is  regarded 
as  satisfactory.  But  for  most  earths  this  would  be  too 
high,  and  accordingly  the  cross-section  of  the  ditch  must 
be  made  wider  and  of  less  depth  in  order  to  make  the  hy- 
draulic radius  smaller  and  thus  diminish  the  velocity. 

The  following  statements  show  approximately  the  veloc- 
ities which  are  required  to  move  different  materials : 

0.25  feet  per  second  moves  fine  clay, 

0.5  feet  per  second  moves  loam  and  earth, 

i.o  feet  per  second  moves  sand, 

2.0  feet  per  second  moves  gravel, 

3.0  feet  per  second  moves  pebbles  i  inch  in  size, 

4.0  feet  per  second  moves  spalls  and  stones, 

6.0  feet  per  second  moves  large  stones. 

The  mean  velocity  in  a  channel  may  be  somewhat  larger 
than  these  values  before  the  materials  will  move,  because 
the  velocities  along  the  wetted  perimeter  are  smaller  than 
the  mean  velocity.  More  will  be  found  on  this  subject  in 
Art.  127. 

Prob.  113.  A  ditch  is  to  discharge  200  cubic  feet  per  second 
with  a  mean  velocity  of  3.4  feet  per  second.  If  its  bottom  width 
is  1 6  feet  and  the  side  slopes  are  i  on  i,  compute  the  depth  of 
water  and  the  slope. 


ART.  114  LARGE  STEEL  AND  WOOD  PIPES  289 


ART.  114.     LARGE  STEEL  AND  WOOD  PIPES 

Long  pipes  of  large  size  are  usually  regarded  as  conduits 
even  when  running  under  pressure,  for  in  formula  (93)  2  the 
ratio  h/l  may  be  replaced  by  the  slope  s  and  the  diameter 
d  is  four  times  the  hydraulic  radius  r  ;  then  it  becomes 


v  —  V^Sg/f^/rs  = 

which  is  the  same  as  the  Chezy  formula.  Values  of  c  may 
be  directly  computed  from  observed  values  of  v,  r,  and  s, 
and  this  has  been  done  by  many  experimenters.  When 
values  of  c  are  known,  all  computations  for  long  pipes  may 
be  made  exactly  like  those  for  circular  conduits. 

By  Herschel's  discussion  of  the  gagings  of  new  steel 
riveted  pipes  made  prior  to  1897  (Art.  101),  the  following 
values  of  the  coefficient  c  were  derived  for  such  pipes  with 
taper  joints: 

Velocity,  feet  per  second,  v=   i  2  3  4  5  6 

for  1.2  feet  diameter,  c=    ........  in  113  116 

for  3.5  feet  diameter,  c=  98       106       109  no  109  109 

for  4.0  feet  diameter,  c=  97       100       102  104  105  105 

and  the  following  are  values  for  cylinder  jointed  pipes: 

Velocity,  feet  per  second,     o>  =    i  2  3  4  5  6 

for  3.0  feet  diameter,  c=   86         95        103       in        117        124 

for  4.0  feet  diameter,  c  =  ioi        109       113        113        112        112 

The  following  values  were  derived  by  Marx,  Wing,  and 
Haskins  from  their  gagings  of  a  6-foot  steel  riveted  pipe 
with  cylinder  joints: 

Velocity,  i>=        I  2  3  4  5 

1897 


j  Chezy,          c=      no  no  108  in 


{ Kutter,       w=o.oi3         0.014         0.015         0.014         

(Chezy,          c=        82  98  102  104  105 

?9(  Kutter,       n=o.oi8         0.016         0.015         0.015         0.015 

and  the  increase  in  roughness  of  the  surface  during  two 
years'  use  is  indicated  by  the  decrease  in  c  or  by  the 
increase  in  n. 


290  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

For  wooden  stave  pipes  the  gagings  of  Noble  and  those 
of  Marx,  Wing,  and  Haskins,  already  referred  to  in  Art.  101, 
furnish  the  following  values  of  the  coefficients  c,  those  given 
for  the  6-foot  diameter  in  the  first  line  being  for  new  pipe, 
and  those  in  the  second  line  after  two  years '  use : 

Velocity,  T>=     i               2  3  4              5 

3.7  feet  diameter,  c=    ...  ...  (109)  113  116 

4.5  feet  diameter,  c=(ii2)  122  126  128 

6.0  feet  diameter,  c=    100  115  122  125  ... 

6. o  feet  diameter,  c=    116  120  121  122  122 

Here  the  two  values  in  parentheses  have  been  found  by  a 
graphic  discussion  of  the  results  of  the  observations.  For 
the  first  of  these  pipes  the  value  of  Kutter's  n  ranges  from 
0.013  to  0.012,  while  for  the  second  and  third  it  is  prac- 
tically constant  at  0.013. 

Prob.  114.  Compute  the  discharge  of  a  new  steel  riveted  pipe 
with  cylinder  joints,  and  also  that  of  a  wooden  pipe,  the  length 
of  each  being  6490  feet,  the  head  37  feet,  and  the  diameter  60 
inches.  Compute  also  the  probable  discharge  after  two  years' 
use. 

ART.  115.     BAZIN'S  FORMULA 

In  1897  Bazin  proposed  a  formula  for  open  channels  as 
the  result  of  an  extended  discussion  of  the  most  reliable 
gagings.*  In  it  the  coefficient  c  is  expressed  in  terms  of 
the  hydraulic  radius  and  the  roughness  of  the  surface,  but 
the  slope  does  not  enter.  It  is 


,    ,- 
0.552  +m/vr 


(115) 


This  is  for  English  measures,  r  being  in  feet  and  v  in  feet 
per  second,  and  the  quantity  m  has  the  following  values  : 

m  =  o.o6  for  smooth  cement  or  matched  boards, 
m  =  o.i6  for  planks  and  bricks, 

*  Annales  des  ponts  et  chaussees,  1897,  4"  trimestre,  pp.  20-70. 


ART.  115  BAZIN'S  FORMULA  291 

m  =  0.46  for  masonry, 
w  =  o.85  for  regular  earth  beds, 
m  =  1.30  for  canals  in  good  order, 
m  =  i .75  for  canals  in  very  bad  order. 

Table  46  gives  values  of  c  computed  from  (115)  for  these 
values  of  m  and  for  several  values  of  r,  from  which  coeffi- 
cients may  be  selected  for  particular  surfaces.  It  may  be 
noted  that  for  a  perfectly  smooth  surface  where  m=o,  the 
formula  gives  v  =  i$&Vrs,  which  cannot  be  correct  since 
uniform  velocity  could  not  exist.  For  this  extreme  case 
Kutter's  formula  appears  to  be  more  satisfactory,  for  if 
n=o  the  value  of  c  is  infinite.  However,  no  empirical 
formula  can  be  tested  by  applying  it  to  an  extreme  case. 

A  comparison  of  the  values  of  c  obtained  from  the 
formulas  of  Kutter  and  Bazin  only  serves  to  emphasize  the 
uncertainty  regarding  the  selection  of  the  proper  coefficient 
in  particular  cases.  Kutter's  n  =  o.oio  corresponds  to 
Bazin's  m=o.i6,  and  for  several  different  hydraulic  radii, 
the  coefficients  for  this  degree  of  roughness  are  as  follows: 

Hydraulic  radius  r  in  feet,  =    i              3              5  7 

From  Bazin's  formula,  0  =  142  149  151  152 

From  Kutter,  s=o.oi,  0  =  156  179  187  196 

From  Kutter,  £=0.001,  0  =  147  178  191  203 

From  Kutter,  ^=0.00005,  0  =  140  178  193  209 

while  the  agreement  is  fair  for  a  hydraulic  radius  of  one  foot 
it  fails  to  be  satisfactory  for  larger  radii.  This  is  perhaps 
a  severe  comparison  because  it  is  probable  that  no  channel 
in  neat  cement  has  ever  been  constructed  having  a  hydraulic 
radius  as  great  as  5  feet,  but  it  serves  to  show  that  these 
empirical  formulas  differ  widely  when  applied  to  unusual 
cases.  For  the  present,  at  least,  the  formula  of  Kutter 
appears  to  receive  the  most  general  acceptance,  but  un- 
doubtedly the  time  will  come  when  it  will  be  replaced  by 
a  more  satisfactory  one.  An  actual  gaging  of  the  discharge 
by  the  method  of  Art.  123  will  always  give  more  reliable 
information  than  can  be  obtained  from  any  formula. 


292  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

For  a  hydraulic  radius  of  3.28  feet  Kutter's  formula  for 
c  reduces  to  the  convenient  expression    • 

i.&n/n        whence        v=  — Vrs 

n 

and  this  may  be  used  for  approximate  computations  when 
r  lies  between  2  and  6  feet.  Here  n  is  the  roughness  factor, 
the  values  of  which  are  given  in  Art.  111.  When  r=3.2& 
feet,  Bazin's  formula  gives  0  =  136.  for  brickwork,  while 
Kutter's  gives  0  =  140;  for  canals  in  good  order  Bazin's 
formula  gives  c  =69,  while  Kutter's  gives  0  =  72.  The  com- 
parison is  very  satisfactory,  and  so  close  an  agreement  is 
not  generally  to  be  expected  when  computations  are  made 
from  different  formulas.  The  formula  of  Bazin  is  largely 
used  in  France  and  England,  and  that  of  Kutter  in  other 
countries. 

Prob.  115a.  Solve  Problem  111  by  the  use  of  Bazin's  coeffi- 
cients.    Also  solve  Problem  1106. 


ART.  116.     OTHER  FORMULAS  FOR  CHANNELS 

Many  attempts  have  been  made  to  express  the  mean 
velocity  and  discharge  in  a  channel  by  the  formulas 

v=Crxsy         q=aCr*sy 

where  x  and  y  are  derived  from  the  data  of  observations 
by  processes  similar  to  those  explained  in  Art.  42.  As  a 
rule  these  attempts  have  not  proved  successful  except  for 
special  classes  of  conduits,  as  the  exponents  of  r  and  s  vary 
with  different  values  of  r  and  with  different  degrees  of 
roughness.  For  conduits  having  the  same  kind  of  surface 
a  formula  of  this  kind  may  be  established  which  will  give 
good  results.  The  values  x  =  f  and  x  =  \  are  frequently 
advocated,  y  being  not  far  from  J;  with  such  values  C  is 
found  to  vary  less  for  certain  classes  than  the  c  of  the  Chezy 
formula,  and  this  is  the  only  argument  in  favor  of  expo- 
nential formulas. 


ART.  116  OTHER  FORMULAS  FOR  CHANNELS  293 

From  their  gagings  of  the  Sudbury  conduit,  Fteley  and 
Stearns  determined  a  formula  for  its  mean  velocity.  The 
section  consists  of  a  part  of  a  circle  of  9.0  feet 
diameter,  having  an  invert  of  13.22  feet 
radius,  whose  span  is  8.3  feet  and  depression 
0.7  feet,  the  axial  depth  of  the  conduit  being 
7.7  feet.  The  conduit  is  lined  with  brick, 
having  cement  joints  one-quarter  of  an  inch 
thick.  The  flow  was  allowed  to  occur  with  different  depths, 
for  each  of  which  the  discharge  was  determined  by  weir 
measurement.  A  discussion  of  the  results  led  to  the  con- 
clusion that  in  the  portion  with  the  brick  lining  the  coeffi- 
cient c  had  the  value  i27r°-12  when  r  is  in  feet,  and  hence 
results  the  exponential  formula 


In  a  portion  of  the  conduit  where  the  brick  lining  was  coated 
with  pure  cement  the  coefficient  was  found  to  be  from  7 
to  8  percent  greater  than  127.  In  another  portion  where 
the  brick  lining  was  covered  with  a  cement  wash  laid  on 
with  a  brush  the  coefficient  was  from  i  to  3  percent  greater. 
For  a  long  tunnel  in  which  the  rock  sides  were  ragged,  but 
with  a  smooth  cement  floor,  it  was  found  to  be  about  40 
percent  less.* 

As  a  sample  of  the  many  exponential  formulas  which 
have  been  advocated,  those  derived  by  Foss  may  be  cited. 
For  surfaces  corresponding  to  Kutter's  values  of  n  less 
than  0.017  he  finds  | 


or         v 
in  which  C  has  the  following  values: 

forn=   0.009      o.oio     o.on      0.012      0.013      0.015      0-017 
(7  =  23  ooo    19  ooo    15  ooo    12  ooo    10  ooo       8000       6000 

*  Transactions  American  Society  Civil  Engineers,  1883,  vol.  12,  p.  114. 
t  Journal  of  Association  of  Engineering  Societies,  1894,  vol.  13,  p.  295. 


294  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

For  surfaces  corresponding  to  Kutter's  values  of  n  greater 
than  0.018,  his  formula  is 


or         v 
and  the  values  of  C  for  this  case  are 


0.025       0.030       0.035 
C=  5000         3000         2000         1500 

For  circular  sections  running  full  he  also  proposes  the  for- 
mula 5  =  o.oo6$q^/d*.  These  formulas  are  open  to  objec- 
tion on  account  of  the  large  range  in  the  values  of  C. 

In  conclusion,  it  may  be  noted  that  when  the  velocity 
is  very  low  the  Chezy  formula  is  not  valid.  In  such  a  case 
the  velocity  does  not  vary  with  the  square  root  of  the  slope, 
but  with  its  first  power,  the  same  conditions  obtaining  as 
in  pipes  (Art.  103).  A  glacier  moving  in  its  bed  at  the 
rate  of  a  few  feet  per  year  has  a  velocity  directly  propor- 
tional to  its  slope.  Water  flowing  in  a  channel  with  a 
velocity  less  than  one-quarter  of  a  foot  per  second  follows 
the  same  law,  and  the  formulas  of  this  chapter  cannot  be 
applied.  The  formula  for  this  case  is  v  =  Cr*s,  but  values 
of  C  are  not  known. 

Prob.  116.  Compute  the  fall  of  the  water  surface  in  a  length 
of  1000  feet  for  a  ditch  where  77  =  3.62  feet  per  second,  7-^2.75 
feet,  and  n  —  0.025;  first,  Foss'  formula,  and  second,  by  formula 
(115)  and  Bazin's  coefficients 


ART.  117.     LOSSES  OF  HEAD 

The  only  loss  of  head  thus  far  considered  is  that  due  to 
friction,  but  other  sources  of  loss  may  often  exist.  As  in 
the  flow  in  pipes,  these  may  be  classified  as  losses  at  en- 
trance, losses  due  to  curvature,  and  losses  caused  by  ob- 
structions in  the  channel  or  by  changes  in  the  area  of 
cross-section. 


ART.  117  LOSSES    OF   HEAD  295 

When  water  is  admitted  to  a  channel  from  a  reservoir  or 
pond  through  a  rectangular  sluice  there  occurs  a  contraction 
similar  to  that  at  the  entrance 
into  a  pipe,  and  which  may  be 
often  observed  in  a  slight  de- 
pression of  the  surface,  as  at  D 
in  the  diagram.  At  this  point, 
therefore,  the  velocity  is  greater  than  the  mean  velocity  v, 
and  a  loss  of  energy  or  head  results  from  the  subsequent 
expansion,  which  is  approximately  measured  by  the  differ- 
ence of  the  depths  d^  and  dy  the  former  being  taken  at  the 
entrance  of  the  channel,  and  the  latter  below  the  depres- 
sion where  the  uniform  flow  is  fully  established.  According 
to  the  experiments  of  Dubuat,  the  loss  of  head  is 

dl  —  d-=m  — 
2g 

in  which  m  ranges  between  o  and  2  according  to  the  con- 
dition of  the  entrance.  If  the  channel  be  small  compared 
with  the  reservoir,  and  both  the  bottom  and  side  edges  of 
the  entrance  be  square,  m  may  be  nearly  2 ;  but  if  these 
edges  be  rounded,  m  may  be  very  small,  particularly  if  the 
bottom  contraction  is  suppressed.  The  remarks  in  Chapter 
V  regarding  suppression  of  the  contractipn  apply  also  here, 
and  it  is  often  important  to  prevent  losses  due  to  contrac- 
tion by  rounding  the  approaches  to  the  entrance.  Screens 
are  sometimes  placed  at  the  entrance  to  a  channel  in  order 
to  keep  out  floating  matter ;  if  the  cross-section  of  the  chan- 
nel is  n  times  that  of  the  meshes  of  the  screen,  the  loss  of 
head,  according  to  (74),  is  (n—  i)v2/2g. 

The  loss  of  head  due  to  bends  or  curves  in  the  channel  is 
small  if  the  curvature  be  slight.  Undoubtedly  every  curve 
offers  a  resistance  to  the  change  in  direction  of  the  velocity, 
and  thus  requires  an  additional  head  to  cause  the  flow  be- 
yond that  needed  to  overcome  the  frictional  resistances. 
Several  formulas  have  been  proposed  to  express  this  loss, 


296  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

but  they  all  seem  unsatisfactory,  and  hence  will  not   be 
presented  here,  particularly  as  the  data  for  determining 

their  constants  are  very  scant.  It 
will  be  plain  that  the  loss  of  head 
due  to  a  curve  increases  with  its 
length  and  decreases  with  its  radius, 
as  in  pipes  (Art.  87).  When  a  chan- 
nel turns  with  a  right  angle,  as  in  Fig. 


FIG.  1176  117^  the  loss  of  head  may 

as  equal  to  the  velocity-head,  since  the  experiments  of  Weis- 
bach  on  such  bends  in  pipes  indicate  that  value.  In  this 
case  there  is  a  contraction  of  the  stream  after  passing  the 
corner  and  the  subsequent  impact  causes  the  loss  of  head. 

The  losses  of  head  caused  by  sudden  enlargement  or  by 
sudden  contraction  of  the  cross-section  of  a  channel  may  be 
estimated  by  the  rules  deduced  in  Arts.  74  and  75.  In  order 
to  avoid  these  losses  changes  of  section  should  be  made 
gradually,  so  that  energy  may  not  be  lost  in  impact.  Ob- 
structions or  submerged  dams  may  be  regarded  as  causing 
sudden  changes  of  section,  and  the  accompanying  losses  of 
head  are  governed  by  similar  laws.  The  numerical  estima- 
tion of  these  losses  will  generally  be  difficult,  but  the  prin- 
ciples which  control  them  will  often  prove  useful  in  arrang- 
ing the  design  of  a  channel  so  that  the  maximum  work  of 
the  water  can  be  rendered  available.  But  as  all  losses  of 
head  are  directly  proportional  to  the  velocity-head  v2/2g,  it 
is  plain  that  they  can  be  rendered  inappreciable  by  giving 
to  the  channel  such  dimensions  as  will  render  the  mean 
velocity  very  small.  This  may  sometimes  be  important  in 
a  short  conduit  or  flume  which  conveys  water  from  a  pond 
or  reservoir  to  a  hydraulic  motor,  particularly  in  cases 
where  the  supply  is  scant,  and  where  all  the  available  head 
is  required  to  be  utilized. 

If  no  losses  of  head  exist  except  that  due  to  friction, 
this  can  be  computed  from  (86)  if  the  velocity  v  and  the 


ART.  117  LOSSES    OF   HEAD  297 

coefficient  c  be  known.  For  since  the  value  of  5  is  v2/c2r 
and  also  h/l,  where  h  is  the  fall  expended  in  overcoming 
friction,  h  may  be  found  from 

h=ls=lv*/c*r  (117) 

but  this  computation  will  usually  be  liable  to  a  large  per- 
centage of  error. 

As  an  example  of  the  computations  which  sometimes 
occur  in  practice  the  following  actual  case  will  be  discussed. 
From  a  canal  A  water  is  carried  through  a  cast-iron  pipe  B 


Tf\      V 


FIG.  117c 

to  an  open  wooden  forebay  C,  where  it  passes  through  the 
orifice  D  and  falls  upon  an  overshot  wheel.  •  At  the  mouth 
of  the  pipe  is  a  screen,  the  area  between  the  meshes  being 
one-half  that  of  the  cross-section  of  the  pipe.  The  pipe 
is  3  feet  in  diameter  and  32  feet  long.  The  forebay  is  of 
unplaned  timber,  5  feet  wide  and  38  feet  long  and  it  has 
three  right-angled  bends.  The  orifice  is  5  inches  deep  and 
40  inches  wride,  with  standard  sharp  edges  on  top  and  sides 
and  contraction  suppressed  on  lower  side  so  that  its  coeffi- 
cient of  contraction  is  about  0.68  and  its  coefficient  of 
velocity  about  0.98.  The  water  level  in  the  canal  being 
3.75  feet  above  the  bottom  of  the  orifice,  it  is  required  to 
find  the  loss  of  head  between  A  and  D. 

The  total  head  0:1  the  center  of  the  orifice  ;s  3.75  —  0.208 
=  3.542  feet.  Let  v1  be  the  mean  velocity  in  the  pipe,  v 
that  in  the  forebay,  and  V  that  in  the  contracted  section 
beyond  the  orifice.  The  area  of  the  cross-section  of  the 
pipe  is  7.07  square  feet;  that  of  the  forebay,  taking  the 
depth  of  water  as  3.7  feet,  is  18.5  square  feet,  and  that  of 
the  contracted  section  of  the  jet  issuing  from  the  orifice 


298  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

is  0.945  square  feet.  It  will  be  convenient  to  express  all 
losses  of  head  in  terms  of  the  velocity-head  v*/2g,  and 
hence  the  first  operation  is  to  express  v^  and  V  in  terms 
of  v,  or  v1  =  2.62V  and  V  =  19.68^.  Starting  with  the  screen, 
the  loss  of  head  due  to  expansion  of  section  after  the  water 
passes  through  it  is,  by  Art.  74, 


The  loss  of  head  in  friction  in  the  pipe,  using  0.02  for  the 
friction  factor  is,  by  Art.  86, 

,,     ,  /  V  vs 

h  =f~r      =i-4  — 
d  2g        *2g 

The  loss  of  head  in  the  expansion  of  section  from  the  pipe 
to  the  forebay  is,  by  Art.  74, 


The  loss  of  head  in  friction  in  the  forebay,  taking  c  from 
Table  46  for  the  hydraulic  radius  1.5  feet  and  the  degree 
of  roughness  w  =  o.i6,  is 


cr 


The  loss  of  head  in  the  three  right-angled  bends  of  the  fore- 
bay  is  estimated,  as  above  noted,  by 


The  loss  of  head  on  the  edges  of  the  orifice  is,  by  Art.  56, 

V2  v2 

hf  =0.041  —  =  15.9  — 

2£          V2g 

Now  the  total  head  is  expended  in  these  lost  heads  and  in 
the  velocity-head  of  the  jet  issuing  from  the  orifice,  or 

v2     V2  v2 

=29.9  --  1  --  =417  — 

V2g        2g         h     '  2g 


ART.  118  VELOCITIES  IN  A  CROSS-SECTION  299 

from  which  the  value  of  v2/2g  is  found  to  be  0.00851  feet. 
Finally  the  total  loss  of  head  before  reaching  the  orifice  is 

v2 
(29.9  —  15.9) —  =  14.0X0.00851  =0.119  feet 

and  therefore  the  water  surface  at  D  is  o.i  19  feet  lower  than 
that  at  A ,  and  the  pressure-head  on  the  center  of  the  orifice 
is  3.433  feet.  This  is  the  result  of  the  computations,  but 
on  making  measurements  with  an  engineer's  level  the  water 
surface  at  D  was  found  to  be  0.125  feet  lower  than  that  at 
A  ;  the  error  of  the  computed  result  is  therefore  0.006  feet. 

Prob.  117.  Compute  from  the  above  data  the  velocities  v, 
v^  and  V  and  the  discharge  through  the  orifice.  Show  that  the 
head  lost  in  passing  through  the  screen  was  0.059  feet,  which  is 
more  than  half  of  the  total. 


ART.  118.     VELOCITIES  IN  A  CROSS-SECTION 

For  a  circular  conduit  running  full  and  under  pressure 
the  velocities  in  different  parts  of  the  section  vary  similarly 
to  those  in  pipes  (Art.  83).  When  it  is  partly  full,  so  that 
the  water  flows  with  a  free  surface,  the  air  resistance  along 
that  surface  is  much  smaller  than  that  along  the  wetted 
perimeter,  and  hence  the  surface  velocities  are  greater  than 
those  near  the  perimeter.  Fig.  118  illustrates  the  varia- 
tion of  velocities  in  a  cross-section  of  the  Sudbury  conduit 
when  the  water  was  about  3  feet  deep,  as  determined  by 
the  gagings  .  of  Fteley  and  Stearns.*  The  97  dots  are 
the  points  at  which  the  velocities  were  measured  by  a  cur- 
rent meter  (Art.  40)  and  the  velocity  for  each  point  in  feet 
per  second  is  recorded  below  it.  From  these  the  contour 
curves  were  drawn  which  show  clearly  the  manner  of  varia- 
tion of  velocity  throughout  this  cross-section.  Since  the 
dots  are  distributed  over  the  area  quite  uniformly,  that  area 
may  be  regarded  as  divided  into  97  equal  parts,  in  each  of 

*  Transactions  American  Society  Civil  Engineers,  1883,  vol.  12,  p.  234. 


300 


FLOW  IN  CONDUITS  AND  CANALS 


CHAP.  IX 


which  the  velocity  is  that  observed,  and  hence  the  mean 
of  the  97  observations  is  the  mean  velocity  (Art.  39).  Thus 
is  found  v  =  2.62o  feet  per  second,  and  this  is  85  percent  of 
the  maximum  observed  velocity. 


SCALE  OF  FEET 


FIG.  118 

If  all  the  filaments  of  a  stream  of  water  in  a  channel 
have  the  same  uniform  velocity  v,  the  kinetic  energy  per 
second  of  the  flow  is  the  weight  of  the  discharge  multiplied 
by  the  velocity-head ;  or 

V2  V2  V3 

K  =  W —  =  wq —  =  wa — 

2g  ^2g  2g 

in  which  W  is  the  weight  of  the  water  delivered  per  second, 
w  is  the  weight  of  one  cubic  unit,  q  the  discharge  per  second, 
and  a  the  area  of  the  cross-section.  For  this  case,  there- 
fore, the  energy  of  the  flow  is  proportional  to  the  area  of 
the  cross-section  and  to  the  cube  of  the  velocity.  Since, 
however,  the  filaments  have  different  velocities  this  expres- 
sion may  be  applied  to  the  actual  flow  by  regarding  v  as 
the  mean  velocity.  To  show  that  this  method  will  be  essen- 
tially correct,  the  above  figure  may  be  discussed,  and  for  it 
the  true  energy  per  second  of  the  flow  is 


_ 


v97*\ 

2g/ 


ART.  119         COMPUTATIONS  IN  METRIC  MEASURES  301 

now  the  ratio  of  this  true  kinetic  energy  to  the  kinetic  energy 
expressed  in  terms  of  the  mean  velocity  is 

Kr  =vl*  +  v23+  .  .  .  +v973 
K  " 


By  cubing  each  individual  velocity  and  also  the  mean  ve- 
locity, there  is  found  Kf  =  0.9992^,  so  that  in  this  instance 
the  two  energies  are  practically  equal,  and  hence  it  is  prob- 
able that  in  most  cases  computations  of  energy  from  mean 
velocity  give  results  essentially  correct. 

Prob.  118.  Draw  a  vertical  plane  through  the  middle  of  Fig. 
118  and  construct  a  longitudinal  vertical  section  showing  the 
distribution  of  velocities.  Also  draw  a  horizontal  plane  through 
the  region  of  maximum  velocity  and  construct  a  longitudinal 
horizontal  section.  Ascertain  whether  the  curves  of  velocity  for 
these  sections  are  best  represented  by  parabolas  or  by  ellipses. 

ART.  119.     COMPUTATIONS  IN  METRIC  MEASURES 

(Art.  106)  The  coefficient  c  in  the  Chezy  formula  de- 
pends upon  the  linear  unit  of  measure.  Let  cx  be  the  value 
when  v  and  r  are  expressed  in  feet  and  c2  the  value  when  v 
and  r  are  expressed  in  meters,  and  let  gl  and  g2  be  the  cor- 
responding values  of  the  acceleration  of  gravity.  Then 
since  c  =  V8g//,  it  is  seen  that 


.  80/32 .16  =0.5520, 

Hence  any  value  of  c  in  the  English  system  may  be  trans- 
formed into  the  corresponding  metric  value  by  multiplying 
by  0.552.  The  metric  value  of  c  for  conduits  and  canals 
usually  lies  between  16  and  100. 

(Art.  107)  Table  38  gives  values  of  the  Chezy  coeffi- 
cient c  for  circular  conduits,  full  or  partly  full.  In  using 
it  a  tentative  method  must  be  employed,  and  for  this  pur- 
pose there  may  be  used  at  first, 

mean  Chezy  coefficient  c  =68 


302  FLOW  IN  CONDUITS  AND  CANALS  CHAP.  DC 

and  then,  after  v  has  been  computed,  a  new  value  of  c  is 
taken  from  the  table  and  a  new  v  is  found.  For  example, 
let  it  be  required  to  find  the  velocity  and  discharge  of  a 
circular  conduit  of  1.5  meters  diameter  when  laid  on  a 
grade  of  0.8  meters  in  1000  meters.  First, 

i)  =  68XiVi-5  X 0.0008  =  1. 1 8  meters  per  second 
and  for  this  velocity  the  table  gives  about  77  for  c.  A 
second  computation  then  gives  ^  =  1.33  meters  per  second 
and  from  the  table  c  is  78.2.  With  this  value  is  found 
^  =  1.35  meters  per  second,  which  maybe  regarded  as  the 
final  result.  When  running  full  the  discharge  of  this  con- 
duit is  o.7854Xi.52Xi-35  =2.39  cubic  meters  per  second. 

(Art.  108)  Table  39  is  the  same  for  all  systems  of 
measures.  The  results  on  page  275,  for  Bazin's  semi- 
circular conduits  of  1.25  meters  diameter  on  a  slope 
5  =0.0015,  are  as  follows,  when  all  dimensions  are  in  meters:. 

For  cement  lining.  For  mortar  lining. 

d'            r             v  C  dr            r            v         c 

0.625     0.314     1.85  85  0.625     0.312      1.69     78 

0.491     0.264     1-61  81  0.515     0.275      1.51      75 

0.314     0.185     1.27  76  0.332     0.194     1.18     69 

0.180     0.112     0.92  71  o.i 86     0.116     0.88     66 

Here  the  coefficient  c  for  any  depth  d'  may  be  roughly  ex- 
pressed by  cl  —  3o(^d  —  d'),  where  cl  is  the  coefficient  for 
the  conduit  when  running  half  full. 

(Art.  109)  Table  41  gives  metric  values  of  c  for  wooden 
and  rectangular  sections  on  a  slope  5  =  0.0049,  as  deter- 
mined by  the  work  of  Darcy  and  Bazin. 

(Art.  110)  In  designing  channels  in  earth  the  following, 
values  may  be  used  for  preliminary  computations: 

for  unplaned  plank,  c  =  55  to  66 

for  smooth  masonry,  c  =  50  to  61 

for  clean  earth,  c  =  33  to  40 

for  stony  earth,      .  c  =  22  to  33 

for  rough  stone,  0  =  19  to  28 

for  earth  foul  with  weeds,  0  =  17  to  28 


ART.  119         COMPUTATIONS  IN  METRIC  MEASURES  303 

(Art.  Ill)     When  r  is  in  meters  and  v  in  meters  per 
second  Kutter's  formula  takes  the  form 

i  0.00155 

-  +  23+- 
n  s 


.ooi 
- 

vV 


in  which  the  number  n  depends  upon  the  roughness  of  the 
surface,  its  values  being  those  given  on  page  282.  It  may 
be  noted  that  when  the  hydraulic  radius  r  is  one  meter, 
the  value  of  c  is  i/n. 

(Art.  112)  Metric  coefficients  for  sewers  will  be  found' 
in  Table  43.  As  these  are  given  to  the  nearest  unit  only, 
the  error  in  using  them  is  slightly  greater  than  with  the 
larger  coefficients  of  the  English  system.  In  important 
cases  the  values  of  c  may  be  directly  computed  from  Kutter's 
formula. 

(Art.  113)  Table  45  in  metric  measures  corresponds  to 
Table  44  in  English  measures  and  is  used  in  the  same  manner. 

(Art.  114)  The  metric  coefficients  c  for  steel  and  wood 
pipes  may  be  obtained  from  those  in  the  text  by  mul- 
tiplying by  0.552,  while  the  velocities  and  diameters  may 
easily  be  replaced  by  metric  equivalents  with  the  help  of 
Table  3. 

(Art.  115)  The  values  of  c  in  Table  47  have  been  taken 
from  the  more  extended  table  published  in  1897  by  Bazin, 
while  those  in  Table  46  have  been  computed  by  (115). 
In  metric  measures  Bazin's  formula  is 


i  +  m/Vr 
in  which  m  has  the  values  given  on  page  290. 

(Art.  116)  The  metric  formula  for  the  Sudbury  conduit 
is  ^  =  8o.9r°-625°-5,  and  Foss'  formula  for  circular  conduits 
or  large  pipes  when  running  full  is  s=o.on8q-"-/d5. 


304  FLOW  IN  CONDUITS  AND  CANALS  CHAP,  ix 

Prob.  119a.  Compute  the  value  of  c  for  a  circular  conduit  1.4 
meters  in  diameter  which  delivers  4.86  cubic  meters  per  second 
when  running  full,  its  slope  being  0.008. 

Prob.  1196.  Find  the  hydraulic  radius  for  a  circular  conduit 
of  1.6  meters  diameter  when  the  water  is  1.2  meters  deep. 

Prob.  119c.  If  the  value  of  c  is  30,  compute  the  depth  of  a 
trapezoidal  section  to  carry  10  cubic  meters  per  second,  the 
slope  s  being  0.0015,  the  bottom  width  double  the  depth,  and 
the  sides  making  an  angle  of  34  degrees  with  the  horizontal. 

Prob.  119d.  A  conduit  lined  with  neat  cement  has  a  cross- 
section  of  3.45  square  meters  and  a  wetted  perimeter  of  5.02 
meters  and  its  slope  is  0.00025.  Compute  the  discharge  in  liters 
per  24  hours,  (a)  by  Kutter's  formula,  and  (6)  by  Bazin's 
formula. 


ART.  120  GENERAL  CONSIDERATIONS  305 


CHAPTER  X 
THE   FLOW   OF   RIVERS 

ART.  120.     GENERAL  CONSIDERATIONS 

Steady  flow  in  a  river  channel  occurs  when  the  same 
quantity  of  water  passes  each  section  in  one  second;  here 
the  mean  velocities  in  different  sections  vary  inversely  as 
the  areas  of  those  sections.  Uniform  flow  is  that  particular 
case  of  steady  flow  where  the  sections  considered  are  equal 
in  area.  Uniform  flow  and  some  other  cases  of  steady  flow 
will  be  mainly  considered  in  this  chapter.  Non-steady  flow 
occurs  when  the  stage  of  a  river  is  rising  or  falling,  and  Art. 
126  treats  of  this  case. 

No  branch  of  hydraulics  has  received  more  detailed 
investigation  than  that  of  the  flow  in  river  channels,  and 
yet  the  subject  is  but  imperfectly  understood.  The  great 
object  of  all  these  investigations  has  been  to  devise  a  sim- 
ple method  of  determining  the  mean  velocity  and  discharge 
without  the  necessity  of  expensive  field  operations.  In  gen- 
eral it  may  be  said  that  this  end  has  not  yet  been  attained, 
even  for  the  case  of  uniform  flow.  Of  the  various  formulas 
proposed  to  represent  the  relation  of  mean  velocity  to  the 
hydraulic  radius  and  the  slope,  none  has  proved  to  be  of 
general  practical  value  except  the  empirical  one  of  Chezy 
given  in  the  last  chapter,  and  this  is  often  inapplicable 
on  account  of  the  difficulty  of  measuring  the  slope  5  and 
determining  the  coefficient  c.  The  fundamental  equations 
for  discussing  the  laws  of  variation  in  the  mean  velocity  v 
and  in  the  discharge  q  are 


where  a  is  the  area  of  the  cross-section  and  r  its  hydraulic 


306  FLOW  OF  RIVERS  CHAP,  x 

radius,  and  all  the  general  principles  of  the  last  chapter  are 
to  be  taken  as  directly  applicable  to  uniform  flow  in  natural 
channels. 

Kutter's  formula  for  the  value  of  c  is  probably  the  best 
in  the  present  state  of  science,  although  it  is  now  generally 
recognized  that  it  gives  too  large  values  for  small  slopes. 
In  using  it  the  coefficients  for  rivers  in  good  condition  may 
be  taken  from  Table  44,  but  for  bad  regimen  n  is  to  be 
taken  at  0.03,  and  for  wild  torrents  at  0.04  or  0.05.  It  is, 
however,  too  much  to  expect  that  a  single  formula  should 
accurately  express  the  mean  velocity  in  small  brooks  and 
large  rivers,  and  the  general  opinion  now  is  that  efforts  to 
establish  such  an  expression  will  not  prove  successful.  In 
the  present  state  of  the  science  no  engineer  can  afford  in 
any  case  of  importance  to  rely  upon  a  formula  to  furnish 
anything  more  than  a  rough  approximation  to  the  discharge 
in  river  channels,  but  actual  field  measurements  of  velocity 
must  be  made. 

When  these  formulas  are  used  to  determine  the  dis- 
charge of  a  river  a  long  straight  portion  or  reach  should  be 
selected  where  the  cross-sections  are  uniform  in  shape  and 
size.  The  width  of  the  stream  is  then  divided  into  a  num- 
ber of  parts  and  soundings  taken  at  each  point  of  division. 
The  data  are  thus  obtained  for  computing  the  area  a  and 
the  wetted  perimeter  p,  from  which  the  hydraulic  depth  r 
is  derived.  To  determine  the  slope  5  a  length  /  is  to  be 
measured,  at  each  end  of  which  bench-marks  are  established 
whose  difference  of  elevation  is  found  by  precise  levels. 
The  elevations  of  the  water  surfaces  below  these  benches 
are  then  to  be  simultaneously  taken,  whence  the  fall  h  in 
the  distance  /  becomes  known.  As  this  fall  is  often  small, 
it  is  very  important  that  every  precaution  be  taken  to 
avoid  error  in  the  measurements,  and  that  a  number  of 
them  be  taken  in  order  to  secure  a  precise  mean.  Care 
should  be  observed  that  the  stage  of  water  is  not  varying 


ART.  121  VELOCITIES  IN  A  CROSS-SECTION  307 

while  these  observations  are  being  made,  and  for  this  and 
other  purposes  a  permanent  gage  board  must  be  established. 
It  is  also  very  important  that  the  points  upon  the  water 
surface  which  are  selected  for  comparison  should  be  situated 
so  as  to  be  free  from  local  influences  such  as  eddies,  since 
these  often  cause  marked  deviations  from  the  normal  sur- 
face of  the  stream.  If  hook  gages  can  be  used  for  referring 
the  water  levels  to  the  benches  probably  the  most  accurate 
results  can  be  obtained.  It  has  been  observed  that  the 
surface  of  a  swiftly  flowing  stream  is  not  a  plane,  but  a 
cylinder,  which  is  concave  to  the  bed,  its  highest  elevation 
being  where  the  velocity  is  greatest,  and  hence  the  two 
points  of  reference  should  be  located  similarly  with  respect 
to  the  axis  of  the  current.  In  spite  of  all  precautions,  how- 
ever, the  relative  error  in  h  will  usually  be  large  in  the  case 
of  slight  slopes,  unless  /  be  very  long,  which  cannot  often 
occur  in  streams  under  conditions  of  uniformity. 

Owing  to  the  uncertainty  of  determinations  of  discharge 
made  in  the  manner  just  described,  the  common  practice  is 
to  gage  the  stream  by  velocity  observations,  to  which  sub- 
ject, therefore,  a  large  part  of  this  chapter  will  be  devoted.- 
The  methods  given  are  equally  applicable  to  conduits  and 
canals,  and  in  Art.  125  will .  be  found  a  summary  which 
briefly  compares  the  various  processes. 

Prob.  120.  Which  has  the  greater  discharge,  a  stream  2  feet 
deep  and  85  feet  wide  on  a  slope  of  i  foot  per  mile,  or  a  stream 
3  feet  deep  and  40  feet  wide  on  a  slope  of  2  feet  per  mile? 

ART.  121.     VELOCITIES  IN  A  CROSS-SECTION 

The  mean  velocity  v  is  the  average  of  all  the  velocities 
of  all  the  small  sections  or  filaments  in  a  cross-section  (Art. 
105).  Some  of  these  individual  velocities  are  much  smaller, 
and  others  materially  larger,  than  the  mean  velocity. 
Along  the  bottom  of  the  stream,  where  the  f fictional  resist- 
•ances  are  the  greatest,  the  velocities  are  the  least;  along 


308 


FLOW  OF  RIVERS 


CHAP.  X 


FIG.  121 


the  center  of  the  stream  they  are  the  greatest.  A  brief 
statement  of  the  general  laws  of  variation  of  these  velocities 
will  now  be  made. 

In  Fig.  121  there  is  shown  at  A  a  cross-section  of  a 
stream  with  contour  curves  of  equal  velocity;  here  the 
greatest  velocity  is  seen  to  be  near  the  deepest  part  of  the 

section  a  short  dis- 
tance below  the  sur- 
face. At  B  is  shown 
a  plan  of  the  stream 
with  arrows  roughly 
representing  the  sur- 
face velocities ;  the 
greatest  of  these  is 
seen  to  be  near  the  deepest  part  of  the  channel,  while  the 
others  diminish  toward  the  banks,  the  curve  showing  the 
law  of  variation  resembling  a  parabola.  At  C  is  shown 
by  arrows  the  variation  of  velocities  in  a  vertical  line,  the 
smallest  being  at  the  bottom  and  the  largest  a  short  dis- 
tance below  the  surface;  concerning  this  curve  there  has. 
been  much  contention,  but  it  is  commonly  thought  to  be 
a  parabola  whose  axis  is  horizontal.  These  are  the  general 
laws  of  the  variation  of  velocity  throughout  the  cross-sec- 
tion; the  particular  relations  are  of  a  complex  character ,. 
and  vary  so  greatly  in  channels  of  different  kinds  that  it 
is  difficult  to  formulate  them,  although  many  attempts 
to  do  so  have  been  made.  Some  of  these  formulas  which 
connect  the  mean  velocity  with  particular  velocities,  such 
as  the  maximum  surface  velocity,  mid-depth  velocity  in 
the  axis  of  the  stream,  etc.,  will  be  given  in  Art.  124. 

Humphreys  and  Abbot  deduced  in  1861  for  the  Missis- 
sippi river*  an  equation  of  the  mean  curve  of  mean  ve- 
locities in  a  vertical  line,  namely, 

V  =  3. 261  -0.7922(^)2 

*  Physics  and  Hydraulics  of  the  Mississippi  River,  edition  of  1876,  p.  243. 


ART.  121  VELOCITIES  IN  A  CROSS-SECTION  309 

in  which  V  is  the  velocity  at  any  distance  y  above  or  below 
the  horizontal  axis  of  the  parabolic  curve  and  d  is  the 
depth  of  the  water,  the  axis  being  at  the  distance  0.297^ 
below  the  surface.  The  depth  of  the  axis  was  found,  how- 
ever, to  vary  greatly  with  the  wind,  an  up-stream  wind 
of  force  4  depressing  it  to  mid-depth  and  a  down-stream 
wind  of  force  5.3  elevating  it  to  the  surface. 

In  a  straight  channel  having  a  bed  of  a  uniform  nature 
the  deepest  part  is  near  the  middle  of  its  width,  while  the 
two  sides  are  approximately  symmetrical.  In  a  river 
bend,  however,  the  deepest  part  is  near  the  outer  bank, 
while  on  the  inner  side  the  water  is  shallow;  the  cause  of 
this  is  undoubtedly  due  to  the  centrifugal  force  of  the  cur- 
rent, which,  resisting  the  change  in  direction,  creates  cur- 
rents which  scour  away  the  outer  bank  or  prevents  deposits 
from  forming  there.  It  is  well  known  to  all  that  rivers 
of  the  least  slope  have  the  most  bends  ;  perhaps  this  is  due 
to  the  greater  relative  influence  of  such  cross  currents 
(see  Art.  147). 

The  theory  of  the  flow  of  water  in  channels,  like  that  of 
flow  in  pipes,  is  based  upon  the  supposition  of  a  mean  ve- 
locity which  is  the  average  of  all  the  parallel  individual  veloc- 
ities in  the  cross-section.  But  in  fact  there  are  numerous 
sinuous  motions  of  particles  from  the  bottom  to  the  surface 
which  also  consume  a  portion  of  the  lost  head.  The  influ- 
ence of  these  sinuosities  is  as  yet  but  little  understood; 
when  in  the  future  this  becomes  known  a  better  theory 
may  be  possible. 

Prob.  121a.  Find  the  approximate  -  discharge  of  a  stream 
whose  width  is  200  feet,  depth  3  feet,  slope  0.6  feet  per  mile, 
when  the  bottom  is  very  stony  and  in  bad  condition. 

Prob.  1216.  Show  that  the  above  formula  for  velocities  in 
a  vertical  can  be  put  into  the  form 


in  which  x  is  the  depth  below  the  surface. 


310  FLOW  OF  RIVERS  CHAP,  x 

ART.  122.     VELOCITY  MEASUREMENTS 

The  method  for  measuring  the  discharge  of  streams 
which  has  been  most  extensively  used  is  by  observing  the 
velocity  of  flow  by  the  help  of  floats.  Of  these  there  are 
three  kinds — surface  floats,  double  floats,  and  rod  floats. 
Surface  floats  should  be  sufficiently  submerged  so  as  to 
thoroughly  partake  of  the  motion  of  the  upper  filaments, 
and  should  be  made  of  such  a  form  as  not  to  readily  be 
affected  by  the  wind.  The  time  of  their  passage  over  a 
given  distance  is  determined  by  two  observers  at  the  ends 
of  a  base  on  shore  by  stop-watches ;  or  only  one  watch  may 
be  used,  the  instant  of  passing  each  section  being  signalled 
to  the  time-keeper.  If  /  be  the  length  of  the  base,  and  t 
the  time  of  passage  in  seconds,  the  velocity  of  the  float  is 
i)  =  l/t.  When  there  are  many  observations  the  numerical 
work  of  division  is  best  done  by  taking  the  reciprocals  of  t 
from  a  table  and  multiplying  them  by  /,  which  for  con- 
venience may  be  an  even  number,  such  as  100  or  200  feet. 

A  sub-surface  float  consists  of  a  small  surface  float 
connected  by  a  fine  cord  or  wire  with  the  large  real  float, 
which  is  weighted  so  as  to  remain  submerged  and  keep 
the  cord  reasonably  taut.  The  surface  float  should  be 
made  of  such  a  form  as  to  offer  but  slight  resistance  to 
the  motion,  while  the  lower  float  is  large,  it  being  the 
object  of  the  combination  to  determine  the  velocity  of  the 
.lower  one  alone.  This  arrangement  has  been  extensively 
used,  but  it  is  probable  that  in  all  cases  the  velocity  of 
the  large  float  is  somewhat  affected  by  that  of  the  upper 
one,  as  well  as  by  the  friction  of  the  cord.  In  general 
the  use  of  these  floats  is  not  to  be  encouraged,  if  any 
other  method  of  measurement  can  be  devised. 

The  rod  float  is  a  hollow  cylinder  of  tin,  which  can 
be  weighted  by  dropping  in  pebbles  or  shot  so  as  to  stand 
vertically  at  any  depth.  When  used  for  velocity  determi- 


ART.  122  VELOCITY   MEASUREMENTS  311 

nations  they  are  weighted  so  as  to  reach  nearly  to  the 
bottom  of  the  channel,  and  the  time  of  passage  over  a 
known  distance  determined  as  above  explained.  It  is 
often  stated  that  the  velocity  of  a  rod  float  is  the  mean 
velocity  of  all  the  filaments  in  contact  with  it.  Theoretically 
this  is  not  the  case,  but  the  rod  moves  a  little  slower. 
However,  in  practice  a  rod  cannot  reach  quite  to  the  bed 
of  the  stream,  and  Francis  has  deduced  the  following 
empirical  formula  for  rinding  the  mean  velocity  Vm  of 
all  the  filaments  between  the  surface  and  the  bed  from 
the  observed  velocity  Vr  of  the  rod : 

Vm  =  Vr(i.oi2-o.ii6\/dr/d) 

in  which  d  is  the  total  depth  of  the  stream  and  df  the 
depth  of  water  below  the  bottom  of  the  rod.*  This  ex- 
pression is  probably  not  a  valid  one,  unless  df  is  less  than 
about  one-quarter  of  d\  usually  it  will  be  best  to  have 
d'  as  small  as  the  character  of  the  bed  of  the  channel 
will  allow. 

The  log  used  by  seamen  for  ascertaining  the  speed 
of  vessels  may  be  often  conveniently  used  as  a  surface 
float  when  rough  determinations  only  are  required,  it 
being  thrown  from  a  boat  or  bridge.  The  cord  of  course 
must  be  previously  stretched  when  wet,  so  that  its  length 
may  not  be  altered  by  the  immersion;  if  graduated  by 
tags  or  knots  in  divisions  of  six  feet,  the  log  may  be  allowed 
to  float  for  one  minute,  and  then  the  number  of  divisions 
run  out  in  this  time  will  be  ten  times  the  velocity  in  feet 
per  second. 

The  determination  of  particular  velocities  in  streams 
by  means  of  floats  appears  to  be  simple,  but  in  practice 
many  uncertainties  are  found  to  arise,  owing  to  wind, 
eddies,  local  currents,  etc.,  so  that  a  number  of  observations 
are  generally  required  to  obtain  a  precise  mean  result. 

*  Lowell  Hydraulic  Experiments,  4th  Edition,  p.  195. 


312  FLOW  OF  RIVERS  CHAP,  x 

For  conduits,  canals,  and  for  many  rivers  the  use  of  a 
current  meter  will  often  be  found  to  be  more  satisfactory 
and  less  expensive  if  many  observations  are  required. 

The  current  meter,  described  in  Art.  40,  is  generally 
operated  from  a  bridge  in  the  case  of  a  small  stream, 
but  it  must  be  often  operated  from  an  anchored  boat 
in  large  rivers.  In  the  latter  case  precise  measurements 
of  surface  velocities  may  be  difficult  on  account  of  the 
eddies  around  the  boat.  Even  when  operated  from  a 
bridge  successful  operation  is  not  easy  when  the  velocity 
exceeds  4  or  5  feet  per  second,  and  special  expedients  are 
necessary  to  keep  the  meter  in  position.  However,  the 
current  meter,  accurately  rated,  will  in  general  do  better 
work  than  can  be  done  by  floats. 

Other  current  indicators  less  satisfactory  for  work  in 
streams  are  the  Pitot  tube  and  the  hydrometric  pendulum, 
shown  in  Fig.  122.  The  former  has  not  been  found  valua- 
ble for  river  measurements,  although  it  has  proved  to  be 
an  instrument  of  great  precision  for  other  classes  of  work 

(Art.    41),    and    the    latter,    al- 
though used  by  some  of  the  early 
hydraulicians,  has  long  been  dis- 
:   carded  as  giving  only  rough  in- 
5     -  dications.     The     same    may    be 

said  of  the  hydrometric  balance, 
FIG.  122  .  .  . 

in    which    weights    measure    the 

intensity  of  the  pressure  of  the  current,  and  of  the  torsion 
balance,  in  which  the  pressure  of  the  current  on  a  sub- 
merged plate  causes  the  tightening  of  a  spring.  These 
instruments  were  used  only  for  measurements  of  velocities 
in  small  channels,  and  they  are  now  mere  curiosities. 

Prob.  122.  A  rod  float  runs  a  distance  of  100  feet  in  42  sec- 
onds, the  depth  of  the  stream  being  6  feet,  while  the  foot  of  the 
rod  is  6  inches  above  the  bottom.  Compute  the  mean  velocity 
in  the  vertical. 


ART.  123  GAGING  AND   DISCHARGE  313 


ART.  123.     GAGING  THE  DISCHARGE 

For  a  very  small  stream  the  most  precise  method  of 
finding  the  discharge  is  by  means  of  a  weir  constructed 
for  that  purpose.  Streams  of  considerable  size  often 
have  dams  built  across  them,  and  these  may  also  be  used 
like  weirs  with  the  help  of  the  coefficients  given  in  Art. 
68,  if  there  be  no  leakage  through  the  dam.  When  there 
are  no  dams  the  method  now  to  be  explained  is  generally 
employed.  In  all  cases  the  first  step  should  be  to  set  up 
a  board  gage,  graduated  to  feet  and  tenths,  and  locate 
its  zero  with  respect  to  the  datum  plane  used  in  the  vicinity, 
so  that  the  stage  of  water  may  at  any  time  be  determined 
by  reading  the  gage. 

The  place  selected  for  the  gaging  should  be  one  where 
the  flow  is  uniform.  One  or  more  sections  at  right  angles 
to  the  direction  of  the  current  are  to  be  established,  and 
soundings  taken  at  intervals  across  the  stream  upon  them, 
the  water  gage  being  read  while  this  is  done.  The  dis- 
tances between  the  places 
of  soundings  are  meas- 
ured either  upon  a  cord 
stretched  across  the  stream 
or  by  other  methods  known 

to  surveyors.  The  data  are  thus  obtained  for  obtaining 
the  areas  av  a2,  a3,  etc.,  shown  upon  Fig.  123,  and  the 
sum  of  these  is  the  total  area  a.  Levels  should  be  run 
out  upon  the  bank  beyond  the  water's  edge,  so  that  in 
case  of  a  rise  of  the  stream  the  additional  areas  can  be 
deduced.  If  a  current  meter  is  used,  but  one  section 
is  needed;  if  floats  are  used,  at  least  two  are  required, 
and  these  must  be  located  at  a  place  wiiere  the  channel 
is  of  as  uniform  size  as  possible. 

The  mean  velocities  vv  v2,  vz,  etc.,  are  next  to  be  de- 
termined for  each  of  the  sub-areas.  If  a  current  meter 


314  FLOW  OF  RIVERS  CHAP,  x 

is  used,  this  may  be  done  by  starting  at  one  side  of  a 
subdivision,  and  lowering  it  at  a  uniform  rate  until  the 
bottom  is  nearly  reached,  then  moving  it  a  few  feet  hori- 
zontally and  raising  it  to  the  surface,  then  moving  it  a 
few  feet  horizontally  and  lowering  it,  and  thus  continuing 
until  the  sub-area  has  been  covered.  The  velocity  then 
deduced  from  the  whole  number  of  revolutions  during 
the  time  of  immersion  is  the  mean  velocity  for  the  sub- 
area.  Or,  the  meter  may  be  simply  raised  and  lowered 
in  a  vertical  at  the  middle  of  the  sub-area,  and  the  result 
will  be  a  close  approximation  to  the  mean  velocity  in 
the  sub-area;  this  in  fact  is  the  usual  method  employed, 
the  division  lines  of  the  sub-areas  being  taken  as  near  to- 
gether as  5  or  10  feet.  When  rod  floats  are  used  they 
are  started  above  the  upper  section,  and  the  times  of 
passing  to  the  lower  one  noted,  as  explained  in  Art. 
122,  the  velocity  deduced  from  a  float  at  the  middle  of 
a  sub-area  being  taken  as  the  mean  for  that  area.  It  will 
be  found  that  the  rod  floats  are  more  or  less  affected  by 
wind,  the  direction  and  intensity  of  which  should  always 
be  noted. 

The  discharge  of  the  stream  is  the  sum  of  the  discharges 
through  the  several  sub-areas,  or 

q  =  ajil  +  a2v2  +  a3v3  +  etc. 

and  if  this  be  divided  by  the  total  area  a,  the  mean  velocity 
for  the  entire  section  is  determined. 

The  following  notes  give  the  details  of  a  gaging  of  the 
Lehigh  River,  near  Bethlehem,  Pa.,  made  at  low  water  in 
1885  by  the  use  of  rod  floats.  The  two  sections  were  100 
feet  apart,  and  each  was  divided  into  10  divisions  of  30  feet 
width.  In  the  second  column  are  given  the  soundings  in 
feet  taken  at  the  upper  section,  in  the  third  the  mean  of 
the  two  areas  in  square  feet,  in  the  fourth  the  times  of 
passage  of  the  floats  in  seconds,  in  the  fifth  the  velocities 


ART.  123  GAGING    AND    DISCHARGE  315 

in  feet  per  second,  which  were  obtained  by  dividing  100 
feet  by  the  times,  and  in  the  last  are  the  products  a^,  a2v2, 
which  are  the  discharges  for  the  subdivisions  av  a2,  etc. 
The  total  discharge  is  found  to  be  826  cubic  feet  per  second, 


Subdivisions 

Depths 

Areas 

Times 

Velocities 

Discharges 

I 

0.0 

55-5 

380 

0.263 

I4.6 

2 

3-0 

148.5 

220 

0-454 

67.4 

3 

6.0 

201  .  7 

185 

0.450 

108.9 

4 

7-1 

217-5 

I  2O 

0.833 

.181.2 

5 

7.0 

2IO.O 

145 

0.690 

144.9 

6 

7.0 

186.0 

150 

0.667 

124.  1 

7 

5-3 

150.8 

I65 

0.606 

91.4 

8 

4-3 

II4.0 

2OO 

0.500 

57-0 

9 

3-° 

2.2 

84.0 

320 

0.313 

26.3 

10 

0.0 

42.O 

430 

0.233 

9-8 

0  =  1410.0  5=825.6 

and  the  mean  velocity  is  v  =  &2  6/1410=0. 59  feet  per  second. 
A  second  gaging  of  the  stream,  made  a  week  later,  when 
the  water  level  was  0.59  feet  higher,  gave  for  the  discharge 
1336  cubic  feet  per  second,  for  the  total  area  1630  square 
feet,  and  for  the  mean  velocity  0.82  feet  per  second. 

As  to  the  accuracy  of  the  above  method,  it  may  be  said 
that  with  ordinary  work,  using  rod  floats,  the  discrepancies 
in  results  obtained  under  different  conditions  ought  not 
to  exceed  10  per  cent;  and  with  careful  work,  using  current 
meters,  they  may  often  be  of  a  higher  degree  of  precision. 
In  any  event  the  results  derived  t  from  such  gagings  are 
more  reliable  than  can  be  obtained  by  any  formula. 

Prob.  123a.  Compute  the  mean  depth  and  the  hydraulic  radius 
for  the  above  section  of  the  Lehigh  River.  Compute  also  the 
value  of  the  coefficient  c  in  the  formula  v  =  cVW,  taking  the 
slope  as  i  on  7500. 

Prob.  1236.  A  stream  140  feet  wide  is  divided  into  seven 
equal  parts,  the  six  soundings  being  1.9,  4.0,  4.8,  4.6,  2.7,  and 
i.o  feet.  The  seven  velocities,  as  found  by  a  current  meter 
are  0.7,  1.6,  2.4,  3.5,  3.0, 1.4,  and  0.6  feet  per  second.  Compute 
the  discharge. 


316  FLOW  or  RIVERS  CHAP,  x 


ART.  124.     APPROXIMATE  GAGINGS 

If  by  any  means  the  mean  velocity  v  of  a  stream  can  be 
found,  the  discharge  is  known  from  the  relation  q=av,  the 
area  a  being  measured  as  explained  in  the  last  article.  An 
approximate  value  of  v  may  be  ascertained  by  one  or  more 
float  measurements  by  means  of  relations  between  it  and 
the  observed  velocity  of  the  floats  which  have  been  deduced 
by  the  discussion  of  observations.  Such  measurements  are 
always  less  expensive  than  those  explained  in  Art.  123, 
and  often  give  information  sufficient  for  the  inquiry  in  hand. 

The  ratio  of  the  mean  velocity  v  to  the  maximum  Surface 
velocity  V  has  been  found  to  usually  lie  between  0.7  and 
0.85,  and  about  0.8  appears  to  be  a  rough  mean  value. 
Accordingly, 

v  =  o.SV] 

from  which,  if  V  be  accurately  determined,  v  can  be  com- 
puted with  an  uncertainty  usually  less  than  20  percent 
Many  attempts  have  been  made  to  deduce  a  more  reliable 
relation  between  v  and  V.  The  following  rule  derived 
from  the  investigations  of  Bazin  makes  the  relation  de- 
pendent on  the  coefficient  c,  the  value  of  which  for  the  par- 
ticular stream  tinder  consideration  is  to  be  obtained  from 
the  evidence  presented  in  the  last  chapter: 


It  is  probable,  however,  that  the  relation  depends  more  on 
the  hydraulic  radius  and  the  shape  of  the  section  than  upon 
the  degree  of  roughness  of  the  channel,  which  c  mainly 
represents. 

The  ratio  of  the  mean  velocity  v±  in  any  vertical  to  its 
surface  velocity  V1  is  less  variable,  for  it  is  found  to  lie 
between  0.85  and  0.92,  or 


ART.  124  APPKOXIMATE    GAGINGS  317 

may  be  used  with  but  an  uncertainty  of  a  few  percent.  If 
several  velocities  Vlt  V2,  etc.,  be  determined  by  surface 
floats,  the  mean  velocities  vv  v2,  etc.,  for  the  several  sub- 
areas  alf  a2,  etc.,  are  known,  and  the  discharge  is  q=a1vl  + 
a2v2  +  etc.,  as  before  explained.  This  method  will  usually 
prove  unsatisfactory  as  compared  with  the  use  of  rod  floats. 

Since  the  maximum  surface  velocity  is  greater  than  the 
mean  velocity  v,  and  since  the  velocities  at  the  shores  are 
usually  small,  it  follows  that  there  are  in  the  surface  two 
points  at  which  the  velocity  is  equal  to  v.  If  by  any  means 
the  location  of  either  of  these  could  be  discovered,  a  single 
velocity  observation  would  give  directly  the  value  of  v. 
The  position  of  these  points  is  subject  to  so  much  variation 
in  channels  of  different  forms,  that  no  satisfactory 'met  hod 
>of  locating  them  has  yet  been  devised. 

The  influence  of  wind  upon  the  surface  velocities  is  so 
great  that  these  methods  of  determining  v  may  not  give 
good  results  except  in  calm  weather.  A  wind  blowing  up 
stream  decreases  the  surface  velocities,  and  one  blowing 
down  stream  increases  them,  without  materially  affecting 
the  mean  velocity  and  discharge. 

By  means  of  a  sub-surface  float,  or  by  a  current  meter, 
the  velocity  V  at  mid-depth  in  any  vertical  may  be  meas- 
ured. The  mean  velocity  v^  in  that  vertical  is  very  closely 

1^=0.987' 

In  this  manner  the  mean  velocities  in  several  verticals  across 
the  stream  may  be  determined  by  a  single  observation  at  each 
point,  and  these  may  be  used,  as  in  Art.  123,  in  connection 
with  the  corresponding  areas  to  compute  the  discharge. 

It  was  shown  by  the  observations  of  Humphreys  and 
Abbot  on  the  Mississippi  that  the'  velocity  V  is  practically 
unaffected  by  wind,  the  vertical  velocity  curves  for  different 
intensities  of  wind  intersecting*  each  other  at  mid-depth. 
The  mid-depth  velocity  is  therefore  a  reliable  quantity  to 


318  FLOW  OF  RIVERS  CHAP,  x 

determine  and  use,  particularly  as  the  corresponding  mean 
velocity  vl  for  the  vertical  rarely  varies  more  than  i  or  2 
percent  from  the  value  0.98^'. 

Prob.  124.  A  stream  60  feet  wide  is  divided  into  three  sec- 
tions, having  the  areas  32,  65,  and  38  square  feet,  and  the  surface 
velocities  near  the  middle  of  these  are  found  to  be  1.3,  2.6,  and 
1.4  per  second.  What  is  the  approximate  mean  velocity  of  the 
stream  and  its  discharge? 

ART.  125.     COMPARISON  OF  METHODS 

This  chapter,  together  with  those  preceding,  furnishes 
many  methods  by  which  the  quantity  of  water  flowing 
through  an  orifice,  pipe  or  channel,  may  be  determined. 
A  few  remarks  will  now  be  made  by  way  of  summary 
and  comparison. 

The  method  of  direct  measurement  in  a  tank  is  always 
the  most  accurate,  but  except  for  small  quantities  is 
expensive,  and  for  large  quantities  is  impracticable.  Next 
in  reliability  and  convenience  come  the  methods  of  gaging 
by  orifices  and  weirs.  An  orifice  one  foot  square  under 
a  head  of  25  feet  will  discharge  about  40  cubic  feet  per 
second,  which  is  as  large  a  quantity  as  can  be  usually 
profitably  passed  through  a  single  opening.  A  weir  20 
feet  long  with  a  depth  of  2.0  feet  will  discharge  about 
200  cubic  feet  per  second,  which  may  be  taken  as  the 
maximum  quantity  that  can  be  conveniently  thus  gaged. 
The  number  of  weirs  may  be  indeed  multiplied  for  larger 
discharges,  but  this  is  usually  forbidden  by  the  expense  of 
construction.  Hence  for  larger  quantities  of  water  indi- 
rect measurements  must  be  adopted. 

The  formulas  deduced  for  the  flow  in  pipes  and  channels 
in  Chaps.  VIII  and  IX  enable  an  approximate  estimation 
of  their  discharge  to  be  determined  when  the  coefficients 
and  data  which  they  contain  can  be  closely  determined. 
The  remarks  in  Art.  120  indicate  the  difficulty  of  ascertain- 


ART.  125          COMPARISON  OF  METHODS  319 

ing  these  data  for  streams,  and  show  that  the  value  of 
the  formulas  lies  in  their  use  in  cases  of  investigation 
and  design  rather  than  for  precise  gagings.  For  pipes  an 
accurately  rated  water  meter  is  a  convenient  method  of 
measuring  the  discharge,  while  for  conduits  it  will  often 
be  found  difficult  to  devise  an  accurate  and  economical 
plan  for  precise  determinations,  unless  the  conditions 
are  such  that  the  discharge  may  be  made  to  pass  over  a 
weir  or  to  be  retained  in  a  large  reservoir  the  capacity 
of  which  is  known  for  every  tenth  of  a  foot  in  depth. 
For  large  aqueducts,  and  for  canals  and  streams,  the 
only  available  methods  are  those  explained  in  this  chapter. 

Surface  floats  are  not  to  be  recommended  except  for  rude 
determinations,  because  they  are  affected  by  wind  and 
because  the  deduction  of  mean  velocities  from  them  is 
always  subject  to  much  uncertainty.  Nevertheless  many 
cases  arise  in  practice  where-  the  results  found  by  the  use 
of  surface  floats  are  sufficiently  precise  to  give  valuable 
information  concerning  the  flow  of  streams.  The  double 
float  for  sub-surface  velocities  is  used  in -deep  and  rapid 
rivers,  where  a  current  meter  cannot  be  well  operated 
on  account  of  the  difficulty  of  anchoring  a  boat.  In 
addition  to  its  disadvantages  already  mentioned  may  be 
noted  that  of  expense,  which  becomes  large  when  many 
observations  are  to  be  taken. 

The  method  of  determining  the  mean  velocities  in 
vertical  planes  by  rod  floats  is  very  convenient  in  canals 
and  channels  which  are  not  too  deep  or  too  shallow.  The 
precision  of  a  velocity  determination  by  a  rod  float  is 
always  much  greater  than  that  of  one  taken  by  the  double 
float,  so  that  the  former  is  to  be  preferred  when  circum- 
stances will  allow.  In  cases  where  the  velocity  is  rapid, 
or  where  there  are  no  bridges  over  the  stream,  rod  floats 
may  often  give  results  more  reliable  than  can  be  obtained 
by  any  other  method. 


320  FLOW  or  RIVERS  CHAP,  x 

Current-meter  observations  are  those  which  now 
generally  take  the  highest  rank  for  precision  in  cases 
where  the  conditions  are  not  abnormal.  The  first  cost 
of  the  outfit  is  greater  than  that  required  for  rod  floats, 
but  if  much  work  is  to  be  done  it  will  prove  the  cheaper. 
The  main  objection  is  to  the  errors  which  may  be  intro- 
duced from  the  lack  of  proper  rating:  this  is  required 
to  be  done  at  intervals,  as  it  is  found  that  the  relation 
between  the  velocity  and  the  recorded  number  of  revo- 
lutions sometimes  changes  during  use. 

In  the  execution  of  hydraulic  operations  which  involve 
the  measurement  of  water  a  method  is  to  be  selected 
which  will  give  the  highest  degree  of  precision  with 
given  expenditure,  or  which  will  secure  a  given  degree 
of  precision  at  a  minimum  expense.  Any  one  can  build 
a  road,  or  a  water-supply  system;  but  the  art  of  engineer- 
ing teaches  how  to  build  it  well,  and  at  the  least  cost  of 
construction  and  maintenance.  Similarly  the  science  of 
hydraulics  teaches  the  laws  of  flow  and  records  the  results 
of  experiments,  so  that  when  the  discharge  of  a  conduit 
is  to  be  measured  or  a  stream  is  to  be  gaged  the  engineer 
may  select  that  method  which  will  furnish  the  required 
information  in  the  most  satisfactory  manner  and  at  the 
least  expense. 

Prob.  125.  Consult  Humphreys  and  Abbot's  Physics  and 
Hydraulics  of  the  Mississippi  River  (Washington,  1862  and 
1876),  and  find  two  methods  of  measuring  the  velocity  of  a 
current  different  from  those  described  in  the  preceding  pages. 

ART.  126.     VARIATIONS  IN  DISCHARGE 

When  the  stage  of  water  rises  and  falls  a  corresponding 
increase  or  decrease  occurs  in  the  velocity  and  discharge. 
The  relation  of  these  variations  to  the  change  in  depth 
may  be  approximately  ascertained  in  the  following  manner, 
the  slope  of  the  water  surface  being  regarded  as  remaining 


ART.  126  VARIATIONS    IN    DISCHARGE  321 

uniform:     Let  the  stream  be  wide,  so  that  its  hydraulic 
radius  is  nearly  equal  to  the  mean  depth  d\  then 

v  =  cVds  =  cs^cfi 
Differentiating  this  with  respect  to  v  and  d  gives 


Here  the  first  member  is  the  relative  change  in  velocity 
when  the  depth  varies  from  d  to  d  ±  dd,  and  the  equation 
hence  shows  that  the  relative  change  in  velocity  is  one- 
half  the  relative  change  in  depth.  For  example,  a  stream 
3  feet  deep,  and  with  a  mean  velocity  of  4  feet  per  second, 
rises  so  that  the  depth  is  3.3  feet;  then  ^  =  4X^X0.3/3  = 
0.2,  and  the  velocity  becomes  4  +  0.2  =4.2  feet  per  second. 

In  the  same  manner  the  variation  in  discharge  may 
be  found.  Let  b  be  the  breadth  of  the  stream,  then 

q  =  cbdVds  =  tbs^cft 

and  by  differentiating  with  respect  to  q  and  d, 

dq/q  =%dd/d 

Hence  the  relative  change  in  discharge  is  ij  times  that  of 
the  relative  change  in  depth.  This  rule,  like  the  preceding, 
supposes  that  dd  is  very  small,  and  will  not  apply  to  large 
variations  in  depth. 

The  above  conclusions  may  be  expressed  as  follows: 
If  the  mean  depth  changes  i  percent,  the  velocity  changes 
0.5  percent,  and  the  discharge  changes  1.5  percent.  They 
are  only  true  for  streams  with  such  cross-sections  that  the 
hydraulic  radius  may  be  regarded  as  proportional  to  the 
depth,  and, even  for  such  sections  are  only  exact  for  small 
variations  in  d  and  v.  They  also  assume  that  the  slope 
5  remains  the  same  after  the  rise  or  fall  as  before;  this 
will  be  the  case  if  a  condition  of  permanency  is  established, 
but,  as  a  rule,  while  the  stage  of  water  is  rising  the  slope 
is  increasing,  and  while  falling  it  is  decreasing. 


322 


FLOW  OF  RIVERS 


CHAP.  X 


Gages  for  reading  the  stages  of  water  are  now  set  up 
on  many  rivers  and  daily  observations  are  taken.  Such 
a  gage  is  usually  a  vertical  board  graduated  to  feet  and 
tenths  and  set  with  its  zero  below  the  lowest  known  water 
level.  Another  form  is  the  box  and  chain  gage  which 
consists  of  a  box  fastened  on  a  bridge  with  a  graduated 
scale  within  it  and  a  chain  that  can  be  let  down  to  the 
water  level.  Such  observations  of  the  daily  stage  of  a 
river  are  of  great  value  in  planning  engineering  constructions,. 
and  they  are  now  made  at  many  stations  by  the  United 
States  government  through  the  Department  of  Agriculture 
and  the  Geological  Survey  Bureau. 

When  several  measurements  of  the  discharge  of  a 
stream  have  been  made  for  different  stages  of  water  a 
curve  may  be  drawn  to  show  the  law  of  variation  of  dis- 
charge, and  from  this  curve  the  discharge  corresponding 
to  any  given  stage  of  water  may  be  approximately  ascer- 
tained. These  discharge  curves  have  been  determined 
by  the  U.  S.  Geological  Survey  for  many  stations  where 
daily  records  of  the  water  stage  are  kept.*  Fig.  126  shows 


—  '  — 

-__., 

••  — 

—  — 

^*—  - 

.-—  • 

_^- 

i  — 

.—  .  •—  ' 

— 

— 

.  •** 

.  —  —  • 

-~-' 

^^* 

^ 

^ 

/ 

0          5000         10000        15000        20000 

FIG.  126 

the  discharge  curve  of  the  Lehigh  river  at  Bethlehem,, 
Pa.,  the  ordinates  being  the  heights  of  the  water  level 
as  read  on  the  gage,  and  the  abscissas  being  the  discharges 
of  the  river  in  cubic  feet  per  second.  This  is  only  a  part 
of  the  discharge  curve  for  that  river,  as  the  water  has  been 
known 'to  rise  to  22.5  feet  and  the  corresponding  discharge 

*  Water  Supply  and  Irrigation  Papers,  Nos.  35-39,  1900. 


ART.  127  TRANSPORTING    CAPACITY 

is  over  100  ooo  cubic  feet  per  second.  Each  station  on 
a  river  has  its  own  distinctive  discharge  curve,  for  the 
local  topography  determines  the  heights  to  which  the 
water  level  will  rise. 

Prob.  126.  A  stream  of  4  feet  mean  depth  delivers  800  cubic 
feet  per  second.  What  will  be  the  discharge  when  the  depth  is 
decreased  to  3.87  feet?  If  the  stream  is  100  feet  wide,  what 
will  be  the  velocity  when  the  depth  is  4.12  feet? 


ART.  127.     TRANSPORTING  CAPACITY  OF  CURRENTS 

The  fact  that  the  water  of  rapid  streams  transports  large 
quantities  of  earthy  matter,  either  in  suspension  or  by 
rolling  it  along  the  bed  of  the  channel,  is  well  known,  and 
has  already  been  mentioned  in  Art.  114.  It  is  now  to  be 
shown  that  the  diameters  of  bodies  which  can  be  moved 
by  the  pressure  of  a  current  vary  as  the  square  of  its  ve- 
locity, and  that  their  weights  vary  as  the  sixth  power  of 
the  velocity. 

When  water  causes  sand  or  pebbles  to  roll  along  the  bed 
of  a  channel  it  must  exert  a  force  approximately  propor- 
tional to  the  square  of  the  velocity  and  to  the  area  exposed 
(Art.  29),  or  if  d  be  the  diameter  of  the  body  and  C  a  con- 
stant, the  force  required  to  move  it  is 

F  =  CdV 

But  if  motion  just  occurs,  this  force  is  also  proportional  to 
the  weight  of  the  body,  because  the  frictional  resistances  of 
one  body  upon  another  varies  as  the  normal  pressure  or 
weight.  And  as  the  weight  of  a  sphere  varies  as  the  cube 
of  the  diameter,  it  follows  that 

d*=CdW        or        d  =  Cv* 

Now  since  d  varies  as  v2,  the  weight  of  the  body,  which  is 
proportional  to  d3,  must  vary  as  ve ;  which  proves  the  prop- 
osition enunciated  above.  Hence  an  increase  in  velocity 
causes  far  greater  increase  in  transporting  capacity. 


324  FLOW  OF  RIVERS  CHAP,  x 

Since  the  weight  of  sand  and  stones  when  immersed  in 
water  is  only  about  one-half  their  weight  in  air,  the  fric- 
tional  resistances  to  their  motion  are  slight,  and  this  helps 
to  explain  the  circumstance  that  they  are  so  easily  trans- 
ported by  currents  of  moderate  velocity.  It  is  found  by 
observation  that  a  pebble  about  one  inch  in  diameter  is 
rolled  along  the  bed  of  a  channel  when  the  velocity  is  about 
3  J  feet  per  second  ;  hence,  according  to  the  above  theoretical 
deduction,  a  velocity  five  times  as  great,  or  17^  feet  per 
second,  will  carry  along  stones  of  25  inches  diameter.  This 
law  of  the  transporting  capacity  of  flowing  water  is  only 
an  approximate  one,  for  the  recorded  experiments  seem 
to  indicate  that  the  diameters  of  moving  pebbles  on  the 
bed  of  a  channel  do  not  vary  quite  as  rapidly  as  the  square 
of  the  velocity.  The  law,  moreover,  is  applicable  only  to 
bodies  of  similar  shape,  and  cannot  be  used  for  comparing 
round  pebbles  with  flat  spalls. 

The  following  table  gives  the  velocities  on  the  bed  or 
bottom  of  the  channel  which  are  required  to  move  the 
materials  stated.  The  corresponding  approximate  mean 
velocities  in  the  cross-section  given  in  the  last  column  are 
derived  from  the  empirical  formula  deduced  by  Darcy, 


in  which  vf  is  the  bottom  and  v  the  'mean  velocity.  The 
bottom  or  transporting  velocities  were  deduced  by  Dubuat 
from  experiments  in  small  troughs,  and  hence  are  probably 
slightly  less  than  the  velocities  which  would  move  the  same 
materials  in  channels  of  natural  earth. 

Bottom         Mean 
velocity      velocity 

Clay,  fit  for  pottery,  0.3  0.4 

Sand,  size  of  anise-seed,  0.4  0.5 

Gravel,  size  of  peas,  0.6  0.8 

Gravel,  size  of  beans,  1.2  1.6 

Shingle,  about  i  inch  in  diameter,  2.5  3.5 

Angular  stones  ,  about  i  J  inches  ,  3.5  4.5 


ART.  128  INFLUENCE    OF    DAMS   AND    PlERS  325 

The  general  conclusion  to  be  derived  from  these  figures  is 
that  ordinary  small,  loose  earthy  materials  will  be  trans- 
ported or  rolled  along  the  bed  of  a  channel  by  velocities  of 
2  or  3  feet  per  second.  It  is  not  necessarily  to  be  inferred 
that  this  movement  of  the  materials  is  of  an  injurious 
nature  in  streams  with  a  fixed  regimen,  but  in  artificial 
canals  the  subject  is  one  that  demands  close  attention. 
The  velocity  of  the  moving  objects  after  starting  has  been 
found  to  be  usually  less  than  half  that  of  the  current.* 

Prob.  127a.  A  stone  weighing  0.5  pounds  is  moved  by  a  cur- 
rent of  3  feet  per  second ;  what  is  the  weight  of  the  largest  stone 
that  can  be  moved  by  a  current  of  9  feet  per  second? 

Prob.  1276.  In  the  early  history  of  the  earth  the  moon  was 
half  its  present  distance  from  the  earth's  center,  and  the  tides 
were  about  eight  times  as  high  as  at  present.  It  is  supposed  that 
these  tides  rolled  over  the  low  lands  and  moved  great  rocks  from 
place  to  place.  The  greatest  velocity  of  such  a  wave  is  \/gd 
where  d  is  the  depth  of  the  water.  What  is  the  probable  weight 
and  size  of  the  largest  rock  that  such  a  current  could  move? 


ART.  128.     INFLUENCE  OF  DAMS  AND  PIERS 

When  a  dam  is  built  across  a  stream  it  is  often  desired 
to  compute  its  height  so  that  the  water  level  may  stand  at 
a  given  elevation.  Thus,  in  the  figures,  CC  represents  the 


FIG.  128a  FIG.  1286 

surface  of  the  stream  before  the  construction  of  the  dam, 
the  depth  of  the  water  being  D,  and  it  is  required  to  find 
the  height  G  of  the  dam  so  that  the  water  surface  may  be 
raised  the  vertical  distance  d.  There  are  two  cases,  the 

*  Herschel,   on  the  erosive  and  abrading  power  of  water,  in  Journal 
Franklin  Institute,  1878,  vol.  75,  p.  330. 


326  FLOW  OF  RIVERS  CHAP,  x 

first  where  the  crest  is  above  the  original  water  level  CCy 
and  the  second  where  it  is  below  that  level;  in  both  cases 
the  discharge  q  must  be  known  in  order  to  compute  the 
height  of  the  dam. 

When  the  crest  is  not  submerged,  as  in  Fig.  128a,  it  is 
seen  that  the  value  of  G  is  D  +  d  —  H,  where  H  is  the  head 
on  the  crest.  Now  from  Art.  64  the  value  of  q  is  mb  (H  +  i  $h)  $ 
where  b  is  the  length  of  the  crest  and  h  is  the  head  due  to 
velocity  of  approach.  Hence  there  results 


in  which  M  is  to  be  taken  from  Art.  68  or  from  Table  29. 
For  example,  let  the  discharge  be  18  ooo  cubic  feet  per 
second,  and  let  the  width  of  the  stream  above  the  dam  be 
600  feet,  and  the  width  on  the  crest  be  525  feet;  also  let 
D  and  d  be  8.5  and  6.0  feet,  and  let  M  be  taken  as  3.33. 
The  mean  velocity  of  approach  is 

18  ooo 

v  =  7—          -  =  2  .  i  feet  per  second 
600X14-5 

whence  the  velocity-head  is  h  =  o.  0155X2.  i2=o.  07  feet. 
Then  from  the  formula  there  results  £  =  9.9  feet,  which  is 
the  required  height  of  the  dam.  In  many  cases  it  will  be 
unnecessary  to  consider  velocity  of  approach  and  h  may 
be  omitted  from  the  formula  ;  if  this  be  done  for  the  exam- 
ple in  hand  the  value  of  G  is  9.8  feet. 

When  it  is  desired  to  raise  the  water  level  only  a  short 
distance  the  crest  of  the  dam  will  be  submerged.  For  this 
case  Fig.  1286  gives  H  =  D  +  d-G  and  H'=D-G.  By 
inserting  these  heads  in  formula  (66)  2  and  neglecting 
velocity  of  approach,  there  is  found 

G=D+$d-%q/MbVd  (128)2 

Here  the  coefficient  M  lies  between  3.09  and  3.37  depending 
on  the  value  of  the  ratio  H'  /H,  and  as  a  mean  3.1  may  be 
used.  For  example,  let  ^  =  400  cubic  feet  per  second, 


5555 

FIG.  128c 


ART.  128  INFLUENCE    OF    DAMS    AND    PlERS  327 

D=4,  d  =  i,  b  =  $o  feet;  then  G  is  found  to  be  2.95  feet. 
The  value  of  H  is  then  2.05  feet  and  that  of  Hf  is  1.05, 
whence  H'/H  is  0.5  closely,  and  from  Art.  66  the  value 
of  M  is  3.11,  which  indicates  that  the  assumed  value  is 
close  enough.  Accordingly  3.0  feet  may  be  taken  as  the 
height  of  the  submerged  dam. 

When  bridge  piers  are  built  in  a  stream  its  cross-section 
is  diminished  and  the  water  level  up-stream  from  the  piers 
stands  at  a  greater  height  than  before.  The  most  common 
problem  is  to  find  how  high 
the  water  will  rise  when  the 
original  width  B  is  to  be  con- 
tracted to  the  width  b.  Let 
D  be  the  mean  depth  of  the 
water  before  the  building  of 
the  piers,  H  the  rise  in  the 
water  level,  and  q  the  dis- 
charge of  the  stream.  Then  the  discharge  q  may  be  re- 
garded as  consisting  of  two  parts,  first  that  passing  over  a 
weir  of  breadth  B  under  the  head  H ,  and  second  that  pass- 
ing through  the  submerged  orifice  of  breadth  b  and  height 
D  under  the  head  H.  Hence,  from  Arts.  64  and  52, 

cV^($B(H  +  h)*  +  bD(H  +  h)*)  =q  (128), 

in  which  h  is  the  head  due  to  the  velocity  of  approach. 
The  coefficient  of  discharge  c  for  weirs  and  orifices  is  about 
0.6,  but  here  it  is  much  larger,  since  there  is  no  crest.  From 
experiments  by  Weisbach  on  a  small  round  pier,  c  appears 
to  be  over  0.9,  and  from  other  discussions  it  appears  in 
some  cases  to  be  a  little  lower  than  0.8.  Its  value  in  any 
event  depends  upon  the  shape  of  the  piers  and  their  cut- 
waters, and  probably  the  best  that  can  now  be  done  is  to 
take  it  as  0.9  for  piers  with  round  ends  and  at  0.8  for  piers 
with  triangular  cutwaters. 

As  an  example  of  the  determination  of  c,  take  the  case 


328  FLOW  OF  RIVERS  CHAP,  x 

of  a  flood  -in  the  Gungal  River,*  where  £  =  650,  6  =  578, 
and  £>  =  35  feet  and  3  =  477  800  cubic  feet  per  second,  and 
where  it  was  observed  that  the  height  H  was  3.6  feet.  The 
mean  velocity  above  the  piers  was  11=477  800/38.6x650 
=  19.0  feet  per  second,  whence  the  velocity-head  ^  =  5.61 
feet.  Inserting  all  these  data  in  the  formula  and  solving 
for  c,  there  is  found  c  =  0.79.  This  is  an  unusual  case  where 
the  velocity  was  very  high,  and  the  piers  had  sharp  cut- 
waters. 

As  an  example  of  the  determination  of  the  height 
H,  take  the  case  of  a  bridge  over  the  Weser,f  where  B  =  593, 
6  =  315,  .D  =  i6.4  feet  and  3  =  46  550  cubic  feet  per  second. 
As  nothing  is  known  about  the  shape  of  the  piers,  c  may 
taken  as  0.8;  then  formula  (128)3  reduces  to 


from  which  H  +  h  is  found  by  trial  to  be  1.55  feet.  Now, 
assuming  H  as  1.2  feet,  the  mean  velocity  above  the  piers 
is  found  to  be  4.3  feet  per  second,  whence  h  is  0.29  feet. 
Accordingly  #  =  1.55  —  0.29=1.26  feet,  and  with  this 
value  the  velocity  above  the  pier  is  found  to  be  4.44  feet 
per  second,  whence  a  better  value  of  k  is  0.31  feet.  This 
gives  H  =  i.24  feet,  which  may  be  regarded  as  the  final 
result  for  the  height  of  the  backwater. 

Prob.  128a.  A  stream  4  feet  deep  which  delivers  150  cubic 
feet  per  second  is  to  be  dammed  so  as  to  raise  the  water  6  feet 
higher.  Find  the  height  of  the  dam  when  the  length  of  the 
overflow  crest  is  112  feet. 

Prob.  1286.  A  river  940  feet  wide  has  a  mean  depth  of  4.1 
feet  and  a  mean  velocity  of  3.3  feet  per  second.  Ten  piers, 
each  12  feet  wide,  are  to  be  built  in  it.  Compute  the  probable 
rise  of  backwater  caused  by  the  piers.  Compute  also  the  proba- 
ble rise  during  a  flood  which  increases  the  mean  depth  to  18.5, 
feet  and  the  mean  velocity  to  5.8  feet  per  second. 

*  Proceedings  British  Institution  Civil  Engineers,  1868,  vol.  27,  p.  222. 
f  D'Aubuisson's    Treatise  on   Hydraulics,    Bennett's  translation   (New 
York,  1857),  p.  189. 


ART.  129  STEADY    NON-UNIFORM    FLOW  329 


ART.  129.     STEADY  NON-UNIFORM  FLOW 

In  Arts.  105-125  the  slope  of  the  channel,  its  cross- 
section,  and  its  hydraulic  radius  have  been  regarded  as 
constant.  If  these  are  variable  in  different  reaches  of 
the  stream  the  case  is  one  of  non-uniformity  and  this 
will  now  be  discussed.  The  flow  is  still  regarded  as  steady, 
so  that  the  same  quantity  of  water  passes  each  section 
per  second,  but  its  velocity  and  depth  vary  as  the  slope 
and  cross-section  change.  Let  there  be  several  reaches 
lr  /2,  .  .  .  ,  /„,  which  have  the  falls  hlt  hz,  .  .  .  ,  hn,  the  water 
sections  being  at,  a2,  .  .  .  ,  aM,  the  hydraulic  radii  rlt  r2, 
.  .  .  ,rn,  and  the  velocities  vv  v2,  .  .  .  ,  vn.  The  total  fall 
hl+h2  +  .  .  .+hn  is  expressed  by  h.  Now  the  head  corre- 
sponding to  the  mean  velocity  in  the  first  section  is  v^/2g. 
The  theoretic  effective  head  for  the  last  section  is  h  +  v12/2g 
while  the  actual  velocity-head  is  vn2/2g.  The  difference 
of  these  is  the  head  lost  in  friction;  or  by  (117), 


... 

2g        2g         C,\       C2\  Cn\ 

in  which  cx2,  C22,  .  .  .  CM2,  are  the  Chezy  coefficients  for  the 
different  lengths.  Now  let  q  be  the  discharge  per  second; 
then,  since  the  flow  is  steady,  the  mean  velocities  are 


and,  inserting  these  in  the  equation,  it  reduces  to 

* 


which  is  a  fundamental  formula  for  the  discussion  of 
steady  flow  through  non-uniform  channels.  This  formula 
shows  that  the  discharge  q  is  a  consequence  not  only  of 
the  total  fall  h  in  the  entire  length  of  the  channel,!  but 


330  FLOW  OF  RIVERS  CHAP,  x 

also  of  the  dimensions  of  the  various  cross-sections.  The 
assumption  has  been  made  that  a  and  r  are  constant  in 
each  of  the  parts  considered;  this  can  be  realized  by 
taking  the  lengths  lv  12,  .  .  .  ln  sufficiently  short.  If  only 
one  part  be  considered  in  which  a  and  r  are  constant, 
an  and  ax  are  equal,  all  the  terms  but  one  in  the  second  mem- 
ber disappear,  and  the  last  equation  reduces  to  q  =  ca\/rh/l 
which  is  the  Chezy  formula  for  the  discharge  in  a  channel 
of  uniform  cross-section. 

An  important  practical  problem  is  that  where  the 
steady  flow  is  non-uniform  in  a  channel  having  a  bed 
with  constant  slope,  a  condition  which  may  be  caused 
by  an  obstruction  below  the  part  considered  or  by  a  sud- 
den fall  below  it.  Let  at  and  a2  be  the  areas  of  the  two 
sections,  I  their  distance  apart,  and  vl  and  v2  the  mean 
velocities.  Then,  if  a  and  r  be  average  values  of  the  areas 
and  hydraulic  radii  of  the  cross-sections  throughout  the 
length  /,  the  last  formula  becomes 


Now  the  important  problem  is  to  discuss  the  change  in 

depth  between  the  two  sections.  For  this  purpose  let 
in  Fig.  129  be  the  longitudinal  profile  of  the  water 

surface,  let  AJD  be  hori- 
zontal,  and  Af  be  drawn 
parallel  to  the  bed  B<f>v 
The  depths  AlBl  and  A2B2 
are  represented  by  dl  and 
FlG-  129  dv  the  latter  being  taken  as 

the  larger.     Let  i  be  the  constant  slope  of  the  bed  BJ32  ; 

then  DC=il,  and  since  DA2=h  and  A2C  =  d2  —  dv  there  is 

found  for  the  fall  in  the  length  /, 

h=n-(d2-dl) 

Inserting  this  value  of  h  in  the  preceding  equation  and 


ART.  129  STEADY   NON-UNIFORM   FLOW  331 

solving  for  /,  there  is  obtained  the  important  formula 

w-4>  -• 

7  __ 


from  which  the  length  /  corresponding  to  a  change  in 
depth  d2  —  dl  can  be  approximately  computed.  This 
formula  is  the  more  accurate  the  shorter  the  length  /, 
since  then  the  mean  quantities  a  and  r  can  be  obtained 
with  greater  precision,  and  c  is  subject  to  less  variation. 

The  inverse  problem,  to  find  the  change  in  depth  when 
I  is  given  cannot  be  directly  solved  by  this  formula,  be- 
cause the  areas  are  functions  of  the  depths.  Since  d2  —  dt 
is  small  compared  with  either  d1  or  dv  it  is  allowable  to 
regard  d2  as  equal  to  d1  when  they  are  to  be  added  or 
multiplied  together.  Hence 

_i_       i      g2*-aS_d22-d12_(d2  +  dl)(d2-dl)      2(d2-d1) 
a,2     af     a,2a22        b2d,2d22  b2d^  b2d,3 

also  making  a  equal  to  at  and  r  equal  to  d1  in  the  last 
formula,  and  solving  for  d2  —  dv  there  is  found 


S  ^2 

from  which  the  change  in  depth  can  be  computed  when 
all  the  other  quantities  are  given. 

Fig.  129  is  drawn  for  the  case  of  depth  increasing 
down-stream,  but  the  reasoning  is  general  and  the  formulas 
apply  equally  well  when  the  depth  decreases  with  the  fall  of 
the  stream.  In  the  latter  case  the  point  A2  is  below  C,  and 
d2  —  dl  will  be  negative.  As  an  example,  let  it  be  required 
to  determine  the  decrease  in  depth  in  a  rectangular  conduit 
5  feet  wide  and  333  feet  long,  which  is  laid  with  its  bottom 
level,  the  depth  of  water  at  the  entrance  being  maintained 
at  2  feet,  and  the  quantity  supplied  being  20  cubic  feet  per 
second.  Here  /  =  333,  6  =  5,^  =  2,^  =  20,  and  i  =  o.  Taking 


332  FLOW  OF  RIVERS  CHAP,  x 

c  =  89,  and  substituting  all  values  in  the  formula, 
there  is  found  d2  —  d1  =  —o.og  feet;  whence  d2  =  i.gi  feet, 
which  is  to  be  regarded  as  an  approximate  probable  value. 
It  is  likely  that  values  of  d2  —  d1  computed  in  this  manner  are 
liable  to  an  uncertainty  of  15  or  20  percent,  the  longer  the 
distance  /  the  greater  being  the  error  of  the  formula.  In 
strictness  also  c  varies  with  depth,  but  errors  from  this 
course  are  small  when  compared  to  those  arising  in  ascer- 
taining its  value. 

Prob.  129.  Explain  why  formula  (129)3  cannot  be  used  for 
the  above  example  when  the  slope  i  is  o.oi. 

ART.  130.     THE  SURFACE  CURVE 

In  the  case  of  steady  uniform  flow,  in  the  channel  where 
the  bed  has  a  constant  grade,  the  slope  of  the  water  surface 
is  parallel  to  that  of  the  bed,  and  the  longitudinal  profile  of 
the  water  surface  is  a  straight  line.  In  steady  non-uniform 
flow,  however,  the  slope  of  the  water  surface  continually 
varies,  and  the  longitudinal  profile  is  a  curve  whose  nature 
is  now  to  be  investigated.  As  in  the  last  article,  the  width 
of  the  channel  will  be  taken  as  constant,  its  cross-section 
will  be  regarded  as  rectangular,  and  it  will  be  assumed  that 
the  stream  is  wide  compared  to  its  depth,  so  that  the  wetted 
perimeter  may  be  taken  as  equal  to  the  width  and  the  hy- 
draulic radius  equal  to  the  mean  depth  (Art.  105).  These 
assumptions  are  closely  fulfilled  in  many  canals  and  rivers. 

The  last  formula  of  the  preceding  article  is  rigidly  exact 
if  the  sections  at  and  a2  are  consecutive,  so  that  /  be- 
comes dl  and  d2  —  d^  becomes  dd.  Making  these  changes, 

dd     i-2c*b2d3 


dl  ~i-q2/gb2d3 

in  which  d  is  the  depth  of  the  water  at  the  place  considered. 
This  is  the  general  differential  equation  of  the  surface  curve, 


.ART.  130  THE  SURFACE  CURVE  333 

I  being  measured  parallel  to  the  bed  BB,  and  d  upward, 
while  the  angle  whose  tangent  is  the  derivative  dd/dl  is 
also  measured  from  BB. 

To  discuss  this  curve,  let  CC  be  the  water  surface  if  the 
slope  were  uniform,  and  let  D  be  the  depth  of  the  water  in 
the  wide  rectangular  channel.  The  sjope  s  of  the  water 


FIG.  130a  FIG.  1306 

surface  is  here  equal  to  the  slope  i  of  the  bed  of  the  channel, 
and  from  the  Chezy  formula  (106), 

=  av=  cbDVri 


This  value  of  q,  inserted  in  the  differential  equation  of  the 
surface  curve,  reduces  it  to  the  form, 


in  which  d  and  /  are  the  only  variables,  the  former  being  the 
ordinate  and  the  latter  the  abscissa,  measured  parallel  to 
the  bed  BB,  of  any  point  of  the  surface  curve.  The  deriv- 
ative dd/dl  is  the  tangent  of  the  angle  which  the  tangent 
at  any  point  of  the  surface  ctirve  makes  with  the  bed  BB 
or  the  surface  CC. 

First,  suppose  that  D  is  less  than  d,  as  in  Fig.  130a, 
where  A  A  is  the  surface  curve  under  the  non-uniform  flow, 
and  CC  is  the  line  which  the  surface  would  take  in  case  of 
uniform  flow.  The  numerator  of  (130)2  is  then  positive, 
and  the  denominator  is  also  positive,  since  i  is  very  small. 
Hence  3d  is  positive,  and  it  increases  with  d  in  the  direction 
of  the  flow;  going  up-stream  it  decreases  with  dlt  and  the 


334  FLOW  OF  RIVERS  CHAP,  x 

surface  curve  becomes  tangent  to  CC  when  d=D.  This 
form  of  curve  is  that  usually  produced  above  a  dam  ;  it  is 
called  the  "  backwater  curve,"  and  will  be  discussed  in 
detail  in  Art.  131. 

Second,  let  d  be  less  than  D,  as  in  Fig.  1306.  The 
numerator  is  then  negative  and  the-  denominator  positive  ; 
dd  is  accordingly  negative  and  A  A  is  concave  to  the  bed 
BB,  whereas  in  the  former  case  it  was  convex.  This  form 
of  surface  curve  is  produced  when  a  sudden  fall  occurs  in 
the  stream  below  the  point  considered;  it  is  called  the 
"drop-down  curve"  and  will  be  discussed  in  Art.  132. 

Formula  (130)!  may  also  be  put  into  another  form  by 
substituting  for  q  its  value  bdv,  where  v  is  the  mean  velocity 
in  the  cross-section  whose  depth  is  d.  It  thus  becomes 


dl     c2      v2-gd 

and  by  its  discussion  the  same  conclusions  are  derived  as 
before.  When  v  is  equal  to  cVtft  the  inclination  dd/dl 
becomes  zero,  and  the  slope  of  the  water  surface  is  parallel 
to  the  bed  of  the  stream.  When  v  is  less  than  c\/di  the 

i  ^ 

numerator  is  negative,  and  if  the  denominator  is  also  neg- 
ative, the  case  of  Fig.  130a  results.  When  v  is  greater 
than  cVdi  and  the  denominator  is  positive  the  case  of  Fig. 
1306  obtains.  When  v  equals  \/gd,  ttye  value  of  dd/dl  is 
infinity  and  the  water  surface  stands  normal  to  the  bed 
of  the  stream;  this  remarkable  case  can  actually  occur 
in  two  ways  and  they  will  be  discussed  in  Art.  133. 

Prob.  130a.  Let  i=  i/ioo,  c  =  8o,  and  D  =  i  foot.  Compute- 
values  of  dd/dl  for  ^=1.23,  d=i.24,  ^=1.25,  ^=1.26,  and  d  = 
1.27  feet;  then  draw  the  surface  curve. 

Prob.  1306.  Let  the  velocity  of  the  stream  be  20  feet  per 
second,  the  value  of  cbeSo,  and  the  slope  be  i  on  2000.  Com- 
pute values  of  dd/dl  for  depths  of  12.2,  12.3,  12.4,  12.5,  and  12.6 
feet  ;  then  draw  the  surface  curve. 


ART.  131  THE  BACKWATER  CURVE  335 

ART.  131.     THE  BACKWATER  CURVE 

When  a  dam  is  built  across  a  channel  the  water  surface 
is  raised  for  a  long  distance  up-stream.  This  is  a  fruitful 
source  of  contention,  and  accordingly  many  attempts  have 
been  made  to  discuss  it  theoretically,  in  order  to  be  able  to 
compute  the  probable  increase  in  depth  at  various  dis- 
tances back  from  a  proposed  dam.  None  of  these  can  be 
said  to  have  been  successful  except  for  the  simple  case 
where  the  slope  of  the  bed  of  the  channel  is  constant  and 
its  cross-section  such  that  the  width  may  be  regarded  as 
uniform  and  the  hydraulic  radius  be  taken  as  equal  to  the 
depth.  These  conditions  are  closely  fulfilled  for  some 
streams,  and  an  approximate  solution  may  be  made  by  the 
formula  (129)2.  It  is  desirable,  however,  to  obtain  an 
exact  equation  of  the  surface  curve,  so  as  to  secure  a  more 
reliable  method. 

For  this  purpose  take  the  differential  equation  of  the 
surface  curve  given  in  (130)2,  and  let  the  independent  vari- 
able d/D  be  represented  by  x.  Then  it  may  be  put  into 
the  more  convenient  form 


*rT" 

in  which  i  is  the  abscissa  and  Dx  the  ordinate  of  any  point 
of  the  curve.     The  general  integral  of  this  is 


which  is  the  equation  of  the  surface  curve,  C  being  the  con- 
stant of  integration.  To  use  this  let  the  logarithmic  and 
circular  function  in  the  second  parenthesis  of  the  second 
member  be  designated  by  $(%)  or  </>(d/D),  namely, 


1 

-     log,  ~—--     arc  cot  -= 


336  FLOW  OF  RIVERS  CHAP,  x 

Then  the  above  value  of  /  may  be  written 


Now  let  d2  be  the  depth  at  the  dam  and  let  I  be  measured 

up-stream   from   that    point 
to  a  section  where  the  depth 
c     is  dr     Then,  taking  the  in- 
teSral  Between  these   limits 
FIG.  131  the    constant    C   disappears, 

and 


which  is  the  practical  formula  for  use.  In  like  manner  d2 
may  represent  a  depth  at  any  given  section  and  dl  any 
depth  at  the  distance  /  up  the  stream. 

When  d  =  D,  the  depth  of  the  backwater  becomes  equal 
to  that  of  the  previous  uniform  flow,  x  is  unity,  and  hence 
/  is  infinity.  The  slope  CC  of  uniform  flow  is  therefore  an 
asymptote  to  the  backwater  curve.  Accordingly  the  depth 
dl  is  always  greater  than  D,  although  practically  the  differ- 
ence may  be  very  small  for  a  long  distance  /. 

In  the  investigation  of  backwater  problems  there  are 
two  cases :  first,  d2  and  d1  may  be  given  and  /  is  to  be  found ; 
and  second,  I  and  one  of  the  depths  are  given  and  the  other 
depth  is  to  be  found*  To  solve  these  problems  the  values 
of  the  backwater  function  <j>(d/D)  computed  by  Bresse  are 
given  in  Table  48.*  The  argument  of  the  table  is  D/d, 
which,  being  always  less  than  unity,  is  more  convenient 
for  tabular  purposes  than  d/D,  since  the  values  of  the  latter 
range  from  i  to  oo  .  By  the  help  of  Table  48  practical 
problems  may  be  discussed,  and  the  following  examples 
will  illustrate  the  method  of  procedure. 

*  Bresse's  Mecanique  appliques  (Paris,  1868),  vol.  2,  p.  556. 


ART.  131  THE  BACKWATER  CURVE  337 

A  stream  of  5  feet  depth  is  to  be  dammed  so  that  the 
water  shall  be  10  feet  deep  a  short  distance  up-stream  from 
the  dam.  The  uniform  slope  of  its  bed  and  surface  is 
0.000189,  or  a  little  less  than  one  foot  per  mile,  and  its 
channel  is  such  that  the  coefficient  c  is  65.  It  is  required 
to  find  at  what  distance  up-stream  the  depth  of  water  is 
6  feet.  Here  D  =  $,  d2  =  io,  dt=6  feet,  1/2  =  5  291,  and 
c2/g  =  i3i.  Now  D/d2=o.$,  for  which  the  table  gives 
<j)(d2/D)  =0.1318,  and  D/d±  =  0.833,  for  which  the  table  gives 
<t>(dj D)  =0.4792.  These  values  inserted  in  (131) 2  give 

/  =  5291  (10-  6)  +  5(5291-131)  (o-4792  -0.1318) 

from  which  £  =  30  125  feet  =  5. 70  miles.  In  this  case  the 
water  is  raised  one  foot  at  a  distance  5.7  miles  up-stream 
from  the  dam,  in  spite  of  the  fact  that  the  fall  in  the  bed 
of  the  channel  is  nearly  5.7  feet. 

The  inverse  problem,  to  compute  d2  or  dv  when  one  of 
these  and  /  are  given,  can  only  be  solved  by  repeated  trials 
by  the  help  of  Table  48.  For  example,  let  £  =  30  125  feet, 
the  other  data  as  above,  and  let  it  be  required  to  determine 
d2  so  that  dl  shall  be  only  5.2  feet,  or  0.2  feet  greater  than 
the  original  depth  of  5  feet.  Here  1^/^=0.962,  for  which 
the  table  gives  <f>(dJD)  =0.9717.  Then  (131)2  becomes 

30  125=5291(^-5.2)4-25  8oo[o.97i7-<£(</2/£>)] 
which  is  easily  reduced  to  the  simpler  form 
32  566  =  5291^  —  25  Soo<f)(d2/D) 

Values  of  d2  are  now  to  be  assumed  until  one  is  found  that 
satisfies  this  equation.  Let  d2  =  8>  feet,  then  (D/d2)  =0.625 
and,  from  the  table,  <j)(dJD)  =0.2180;  substituting  these, 
the  second  member  becomes  36  703,  which  shows  that  the 
assumed  value  is  too  large.  Again,  take  d2  =  7  feet,  then 
Z)/d2  =  0.714,  for  which  (j)(D/d2)  =0.303^  whence  the  second 
member  is  29  202,  showing  that  7  feet  is  too  small.  If 


338  FLOW  OF  RIVERS  CHAP,  x 


^2  =  7.4  feet,  then  D/d2  =0.675  and  4>(dJD)  =0.2629, 
with  these  values  the  equation  is  nearly  satisfied,  but  7.4 
is  still  too  small.  On  trying  7.5  it  is  found  to  be  too  large. 
The  value  of  d2  hence  lies  between  7.4  and  7.5  feet,  which 
is  as  close  a  solution  as  will  generally  be  required.  The 
height  of  dam  required  to  maintain  this  depth  may  now 
be  computed  from  Art.  128. 

If  the  slope,  width,  or  depth  of  the  stream  changes  ma- 
terially, the  above  method,  in  which  the  distance  /  is  meas- 
ured from  the  dam  as  an  origin,  cannot  be  used.  In  such 
cases  the  stream  should  be  divided  into  reaches,  for  each  of 
which  the  slope,  width,  and  depth  can  be  regarded  as  con- 
stant. The  formula  can  then  be  used  for  the  first  reach  and 
the  depth  of  its  upper  section  be  determined  ;  then  the  appli- 
cation can  be  made  to  the  next  reach,  and  so  on  in  order. 
For  common  rivers  and  for  shallow  canals  it  will  probably  be 
a  good  plan  to  determine  D  by  actual  measurement  of  the 
area  and  wetted  perimeter  of  the  cross-section,  the  hydraulic 
radius  computed  from  these  being  taken  as  the  value  of  D. 
Strictly  speaking  the  coefficient  c  varies  with  the  slope 
and  with  D,  and  its  values  may  be  found  by  Kutter's  for- 
mula, if  it  be  thought  worth  the  while.  Even  if  this  be  done, 
the  results  of  the  computations  must  be  regarded  as  liable 
to  considerable  uncertainty.  In  computing  depths  for 
given  lengths  an  uncertainty  of  10  percent  or  more  in  the 
value  of  d2  —  dv  should  be  expected. 

Prob.  131a.  A  stream,  having  a  cross-section  of  2400  square 
feet  and  a  wetted  perimeter  of  300  feet,  has  a  uniform  slope  of 
2.07  feet  per  mile,  and  its  channel  is  such  that  0  =  70.  It  is 
proposed  to  build  a  dam  to  raise  the  water  6  feet  above  the 
former  level,  without  increasing  the  width.  Compute  the  rise 
of  the  backwater  at  a  distance  of  one  mile  up-stream. 

Prob.  1316.  A  stream  has  the  same  cross-section,  wetted 
perimeter,  slope,  and  coefficient  as  above,  and  the  dam  is  to 
be  built  to  the  same  height.  At  what  distance  up-stream  will 
the  mean  depth  be  10.5  feet? 


ART.  132  THE    DROP-DOWN   SURFACE   CURVE  339 


ART.  132.     THE  DROP-DOWN  SURFACE  CURVE 

When  a  sudden  fall  occurs  in  a  stream,  the  water  surface 
for  a  long  distance  above  it  is  concave  to  the  bed,  as  seen 
in  Fig.  1306  or  in  Fig.  132.  This  case  also  occurs  when  the 
entire  discharge  of  a  canal  is 
allowed  to  flow  out  through  a 
forebay  F  to  supply  a  water- 
power  plant.  Let  D  be  the 
original  uniform  depth  of 

water  having  its  surface  paral- 

lei  to  the  bed,  the  slope  of  both  FIG.  132 

being  i.  Let  dl  and  d2  be  two  of  the  depths  after  the  steady 
non-uniform  flow  has  been  established  by  letting  water  out 
at  F,  and  let  d^  be  greater  than  d2,  the  distance  between 
them  being  /.  The  investigation  of  the  last  article  applies 
in  all  respects  to  this  form  of  surface  curve,  and 


is  the  equation  for  practical  use,  in  which  c  is  the  coefficient 
in  the  Chezy  formula  v  =  cVrs,  and  g  is  the  acceleration  of 
gravity.  Table  48  cannot,  however,  be  used  for  this  case 
because  d/D  in  that  table  is  greater  than  unity,  while  here 
it  is  less  than  unity. 

The  function  $(d/D)  with  values  of  d/D  less  than  unity 
is  here  called  the  "drop-down  function,"  in  order  to  dis- 
tinguish it  from  the  backwater  function  of  the  last  article, 
although  the  algebraic  expression  for  the  two  functions  is 
the  same.  Table  49,  clue  also  to  Bresse,  gives  values  of  this 
drop-down  function  for  values  of  the  argument  d/D  ranging 
from  o  to  i,  and  by  its  use  approximate  solutions  of  prac- 
tical problems  can  be  made.  For  example,  take  a  canal  10 
feet  deep,  having  a  coefficient  c  equal  to  80,  and  let  the 
slope  of  its  bed  be  1/5000  and  its  surface  slope  be  the  same 


340  FLOW  OF  RIVERS  CHAP,  x 

when  the  water  is  in  uniform  flow.  Here  D  =  10  feet,  c2/g  = 
200,  and  1/2  =  5000.  Then 

/-  -5  ooo(dl-d2) 

Now  suppose  that  a  break  occurs  in  the  bank  of  the  canal 
out  of  which  rushes  more  water  than  that  delivered  in  nor- 
mal flow  when  the  depth  is  10  feet,  and  let  it  be  required  to 
find  the  distance  between  two  points  where  the  depths  of 
water  are  8  and  7  feet.  Here  d1  /D  =0.8  for  which  $(dJD) 
=  0.3459,  and  d2/D=o.j  for  which  </>(d2/D)  =0.1711.  In- 
serting these  values  in  the  equation  there  is  found/ =  3390 
feet. 

In  this  case  there  is  a  certain  limiting  depth  below  which 
the  above  formula  is  not  valid.  This  limit  is  the  value  of 
x  for  which  dl/dx  becomes  zero  or  the  value  of  x  where  the 
surface  curve  is  vertical  and  the  bore  occurs  (Art.  133). 
From  (131)1  it  is  seen  that  this  happens  when 

x3  =  c2i/g         or        d=D(c2i/g)* 

and  for  the  above  example  this  limiting  depth  is  found  to  be 
3.4  feet.  Near  this  limit,  however,  the  velocity  becomes 
large,  so  that  there  is  much  uncertainty  regarding  the  value 
of  the  coefficient  c. 

When  a  given  discharge  per  second  is  taken  out  of  a  fore- 
bay  at  the  end  of  a  canal  having  its  bed  on  a  slope  i ,  the 
above  formula  must  be  modified.  Let  q  be  the  discharge 
and  let  Dt  be  the  depth  at  a  section  where  the  slope  is  s; 
then  q  equals  cbD^D^s.  If  this  value  of  q  be  substi- 
tuted in  the  equation  (130)!  and  then  the  same  reasoning 
be  followed  as  at  the  beginning  of  Art.  131,  it  will  be  found 
that  formula  (132)  will  apply  to  this  case  if  D^s/i)*  be 
used  instead  of  D.  For  example,  let  q  =  3000  cubic  feet  per 
second,  Dl  =  io  feet,  i  =  i/io  ooo,  c  =  8o,  and  the  width 
£  =  ioo  feet.  Then 

s=q*/c*b*Dl*  =  1/7100        D  =D1(s/i)*  =  11. 2  feet. 


ART.  132  THE   DROP-DOWN    SURFACE    CURVE  341 

Now  if  it  be  required  to  find  the  distance  between  two 
points  where  the  depths  of  water  are  10  and  9  feet,  formula 
(132)  can  be  directly  applied,  and  accordingly  there  is  found, 
by  the  help  of  Table  49,  " 

/=  —  10  000(10  —  9)  +  109  800(0.578  —  0.355)  =14  400  feet 

and  hence  a  forebay  admitting  the  given  discharge  will  not 
draw  down  the  water  to  a  depth  less  than  9  feet  if  it  be 
located  1  4  400  feet  down-stream  from  the  section  where  the 
mean  depth  is  10  feet. 

Navigation  canals  are  often  built  with  the  bed  horizontal 
between  locks,  and  here  i  =  o.  The  above  formula  cannot 
be  applied  to  this  case  because  the  differential  equation 
(130)  2  vanishes  when  i  is  zero.  To  discuss  it,  equation  (130)4 
must  be  resumed,  and,  inverting  the  same, 

dl      c2W3_c2 
dd~     q*      ~g 

The  integration  of  this  between  the  limits  d^  and  d2  gives 


from  which  /  may  be  computed  when  q  is  known.  As  an 
example,  take  a  rectangular  trough  for  which  q  =  2o  cubic 
feet  per  second,  6  =  5  feet,  0=89,  and  let  0^  =  2.00  feet  and 
d2  =  1.91  feet.  Then  from  the  formula  I  is  found  to  be  329 
feet.  This  is  the  reverse  of  the  example  at  the  end  of  Art. 
129,  where  /  was  given  as  333  feet,  so  that  the  agreement  is 
very  good. 

To  compare  a  canal  having  a  level  bed  with  the  one  pre- 
viously considered,  the  same  data  will  be  used,  namely, 
^  =  10  feet,  d2=9  feet,  b  =  ioo  feet,  c=8o,  and  ^  =  3000 
cubic  feet  per  second.  Then  from  (132)2  there  is  found 

Z  =  i.  778(10*  —  94)  —  200(10  —  9)  =592° 


342  FLOW  OF  RIVERS  CHAP,  x 

and  accordingly  the  water  level  is  drawn  down  in  one-third  of 
the  distance  of  that  of  the  previous  case.  The  quantity  of 
water  that  can  be  obtained  from  a  navigation  canal  is  always 
less  than  from  one  having  a  sloping  bed,  and  it  has  frequently 
happened,  when  such  a  canal  is  abandoned  for  navigation 
purposes  and  is  used  to  furnish  water  for  power  or  for  a 
public  supply,  that  the  quantity  delivered  is  very  much 
smaller  than  was  expected. 

Prob.  132.  A  canal  from  a  river  to  a  power  house  is  two 
miles  long,  its  bed  is  on  a  slope  of  i/io  ooo,  and  c  is  70.  When 
the  water  is  in  uniform  flow  the  depth  D  is  6.0  feet,  and  the 
discharge  is  800  cubic  feet  per  second.  If  there  be  a  power 
house  which  takes  1000  cubic  feet  per  second,  find  the  probable 
depth  of  water  at  the  entrance  to  its  forebay. 


ART.  133.     THE  JUMP  AND  THE  BORE 

A  very  curious  phenomenon  which  sometimes  occurs 
in  shallow  channels  is  that  of  the  so-called  "jump,"  as 
shown  in  Fig.  133a.  This  happens  when  the  denominator 

in  (130)  3  is  zero;  then  dd/dl  is 
infinite,  and  the  water  surface 
^  stands  normal  to  the  bed. 
Placing  that  denominator  equal 
to  zero,  there  is  found  v2=gd. 

Now  by  further  consideration  it  will  appear  that  the  varying 
denominator  in  passing  through  zero  changes  its  sign.  Above 
the  jump  where  the  depth  is  d1  the  velocityjs  slightly  greater 
than  Vgc^  and  below  it  is  less  than  Vgd2.  The  conditions 
for  the  occurrence  of  the  jump  are  that  an  obstruction  should 
be  in  the  stream  below,  that  the  slope  i  should  not  be  small, 
and  that  the  velocity  v1  should  be  greater  than  \/gdr  To 
find  the  necessary  slope,  the  algebraic  conditions  are 


v1=c\/d1i      and     v1^>Vgd1      whence 
and  accordingly  the  jump  cannot  occur  when  i  is  less  than 


ART.  133  THE    JUMP    AND    THE    BORE  343 

g/c2.  For  an  unplaned  planked  trough  c  may  be  taken  at 
about  100;  hence  the  slope  for  this  must  be  equal  to  or 
greater  than  0.00322. 

To  determine  the  height  of  the  jump,  let  d2  —  d^  be  repre- 
sented by  /.  It  is  then  to  be  observed  that  the  lost  velocity- 
head  is  (vlz  —  v22)/2g,  and  that  this  is  lost  in  two  ways  —  first 
by  the  impact  due  to  the  expansion  of  section  (Art.  74), 
and  second  by  the  uplifting  of  the  whole  quantity  of  water 
through  the  height  ^(d2  —  d1),  loss  in  friction  between  dt  and 
d2  being  neglected.  Hence 

V-V^K-pJ2    / 

2g  2g  2 

Inserting  in  this  the  value  of  v2,  found  from  the  relation 
f)  =Vjdlt  and  solving  for  /,  gives 


(133) 

The  following  is  a  comparison  of  heights  of  the  jump  com- 
puted by  this  formula  and  the  observed  values  in  four  ex- 
periments made  by  Bidone,  the  depths  being  in  feet  : 


Depth  di. 

Velocity  /y1. 

Observed  /. 

Computed  /. 

0.149 

4-59 

0.274 

0.290 

o.i54 

4-47 

0.267 

0.283 

0.208 

5-59 

0.305 

0.428 

0.246 

6.28 

0-493 

o.53i 

The  agreement  is  very  fair,  the  computed  values  being  all 
slightly  greater  than  the  observed,  which  should  be  the  case, 
because  the  reasoning  omits  the  frictional  resistances  be- 
tween the  points  where  dl  and  d2  are  measured.  Experi- 
ments made  at  Lehigh  University,  under  velocities  ranging 
from  2.2  to  6.2  feet  per  second,  show  also  a  good  agreement 
between  computed  and  observed  value.*  The  depths  in 
these  experiments  were  less  than  in  those  of  Bidone,  but 
higher  relative  jumps  were  obtained.  For  instance,  for 

*  Engineering  News,  1895,  vol.  34,  p.  28. 


344  FLOW  OF  RIVERS  CHAP,  x 

^=4.33  feet  per  second  and  ^=0.039  feet,  the  observed 
value  of  /  was  0.166  feet,  whereas  the  value  computed  from 
the  above  formula  is  0.173  ^eet'  nere  tne  JumP  is  more  than 
four  times  the  depth  dv  while  it  is  usually  less  than  twice 
dl  in  the  above  records  from  Bidone. 

Another  remarkable  phenomenon  is  that  of  the  so-called 
' '  bore ' '  where  a  tidal  wave  moves  up  a  river  with  a  vertical 
front.  It  is  also  seen  when  a  large  body  of  water  moves 
down  a  canon  after  a  heavy  rainfall,  or  when  a  reservoir 
bursts  and  allows  a  large  discharge  to  suddenly  escape  down 
a  narrow  valley.  In  the  great  flood  of  1889  at  Johnstown, 
Pa.,  such  a  vertical  wall  of  water,  variously  estimated  at 
from  10  to  30  feet  in  height,  was  seen  to  move  clown  the 
valley  carrying  on  its  front  brush  and  logs  mingled  with 
spray  and  foam.*  In  41  minutes  it  travelled  a  distance 'of 
13  miles  down  the  descent  of  380  feet.  The  velocity  was 
hence  about  28  feet  per  second. 

Fig.  1336  shows  the  form  of  surface  curve  for  this  case 
and  by  reference  to  (130)8  it  is  seen  that  dd/dl  must  be  nega- 
tive and  that  it  has  the  value  oo  at  the  vertical  front. 
The  conditions  for  the  occurrence  of  the  bore  then  are 


\/gd     and     z;>c\/5t     whence     i<g/c2 

For  the  Johnstown  flood,  taking  v  as 
28  feet  per  second,  the  value  of  d  found 
w/mw////Mti/z\  from  this  equation  is  24  feet;  it  was 
FIG.  1336  probably    greater   than    this    in    the 

upper  part  of  the  valley  and  less  in  the  lower  part.  Since 
the  value  of  i  is  about  1/180,  it  follows  that  c  must  have 
been  less  than  76.  The  conditions  here  established  show 
that  the  flood  bore  will  occur  when  the  velocity  becomes 
equal  to  Vgd,  provided  c  is  less  than  Vg/i.  It  appears, 
therefore,  that  roughness  of  surface  is  an  essential 'condition 
for  the  formation  of  the  bore  in  a  steep  valley. 

*  Transactions  American  Society  Civil  Engineers,  1889,  vol.  21.  p.  564. 


ART.  133  THE    JUMP   AND    THE    BORE  345 

The  bore  can  also  occur  in  a  canal  with  horizontal  bed 
when  a  lock  breaks  above  an  empty  level  reach,  provided  v 
becomes  equal  to  \^gd.  No  case  of  this  kind  appears  to  be 
on  record,  and  there  seems  to  be  no  way  of  ascertaining 
whether  the  actual  velocity  will  reach  the  limit  Vgd.  If 
the  bore  occurs  and  the  depth  of  the  vertical  wall  be  d2  its 
distance  from  a  point  where  the  depth  is  dl  is  found  from 
(132) 2  by  inserting  in  it  the  value  of  g  corresponding  to  the 
critical  velocity  v.  Thus  may  be  shown  that  for  c  =  80 
and  d2  =^d1  the  length  /  is  about  275^. 

The  tidal  bore,  which  occurs  in  many  large  rivers  when 
the  tide  flows  in  at  their  mouths,  obeys  similar  laws.  Here 
the  slope  i  may  be  taken  as  zero,  while  c  is  probably  very 
large,  so  that  roughness  of  surface  is  not  an  essential  condi- 
tion. The  great  bore  at  Hangchow,  China,  which  occurs 
twice  a  year,  is  said  to  travel  up  the  river  at  a  rate  of  from 
10  to  13  miles  per  hour,  the  height  of  the  vertical  front 
being  from  10  to  20  feet.*  Fromv  =  \/gd,  the  velocity 
corresponding  to  a  depth  of  10  feet  is  12.6  miles  per  hour, 
while  that  corresponding  to  a  depth  of  20  feet  is  17  miles 
per  hour,  so  that  the  statements  have  a  fair  agreement 
with  the  theoretical  law.  This  investigation  indicates  that 
the  velocity  of  the  tidal  bore  depends  mainly  upon  the 
depth  of  the  tidal  wave  above  the  river  surface,  but  it  may 
be  noted  that  other  discussions!  regard  the  depth  of  the 
river  itself  as  an  element  of  importance,  and  Art.  182  con- 
siders this  question  with  respect  to  common  waves. 

Prob.  133a.  When  the  height  of  the  jump  is  three  times  the 
depth  dlt  show  that  the  velocity  vl  must  be  2\/2gdl.  Also  show 
that  0.4146^  is  the  minimum  height  of  a  jump. 

Prob.  1336.  Assuming  that  c  was  32  for  the  valley  of  the 
Johnstown  flood,  and  that  this  was  of  uniform  width,  show  that 

*  Skidmore's  China,  the  Long-lived  Empire  (New  York,  1900),  p.  294, 
•\  G.  H.  Darwin,  The  Tides,  p.  65 ;    Century  Magazine,  vol.  34,  p.  903. 


346  FLOW  OF  RIVERS  CHAP,  x 

the  depth  of  the  vertical  front  at  the  city  was  not  less  than  one- 
half  of  the  depth  a  short  distance  below  the  reservoir. 

Prob.  133c.  Show  that  the  formula  on  page  324,  when  re- 
duced to  the  metric  system,  becomes  v  =  vr  +  6.i^/rs. 

Prob.  133d.  A^  stream  181  meters  wicle  and  5  meters  deep 
has  a  discharge  of  1318  cubic  meters  per  second.  Find  the 
height  of  backwater  when  the  stream  is  contracted  by  piers 
and  abutments  to  a  width  of  96  meters. 

Prob.  133<?.  Which  has  the  greater  discharge,  a  stream  1.2 
meters  deep  and  20  meters  wide  on  a  slope  of  three  meters  per 
kilometer,  or  a  stream  1.6  meters  deep  and  26  meters  wide  on 
a  slope  of  two  meters  per  kilometer? 

Prob.  133/.  A  stream  2  meters  deep  is  to  be  dammed  so  that 
water  shall  be  4  meters  deep  at  the  dam.  Its  slope  is  0.0002  and 
its  channel  is  such  that  the  metric  value  of  c  is  39.  Compute 
the  distance  to  a  section  up-stream  where  the  depth  of  water  is 
3.6  meters. 


ART.  134  RAINFALL   AND    EVAPORATION  347 


CHAPTER  XI 
WATER  SUPPLY  AND  WATER  POWER 

ART.  134.     RAINFALL  AND  EVAPORATION 

All  the  water  that  flows  in  streams  has  at  some  previous 
time  been  precipitated  in  the  form  of  rain  or  snow.  The 
word  "  rainfall"  means  the  total  rain  and  melted  snow,  and 
it  is  usually  measured  in  vertical  inches.  The  annual  rain- 
fall is  least  in  the  frigid  zone  and  greatest  in  the  torrid  zone ; 
at  the  equator  it  is  about  100  inches,  at  latitude  40°  about 
40  inches,  and  at  latitude  60°  about  20  inches.  There  are, 
however,  certain  places  where  the  annual  rainfall  is  as  high 
as  500  inches,  and  others  where  no  rain  ever  falls.  In  the 
United  States  the  heaviest  annual  rainfall  is  near  the  Gulf 
of  Mexico,  where  60  inches  is  sometimes  registered,  and  near 
Puget  Sound,  where  90  inches  is  not  uncommon.  In 
that  large  region,  formerly  called  the  Great  American 
Desert,  which  lies  between  the  Rocky  and  Sierra  Nevada 
mountains,  the  mean  annual  rainfall  does  not  exceed  1 5  inches, 
and  in  Nevada  it  is  only  about  7^  inches.  The  amount  of 
rainfall  in  any  locality  depends  upon  the  winds  and  upon 
the  neighboring  mountains  and  oceans. 

At  any  place  the  rainfall  in  a  given  year  may  vary  con- 
siderably from  the  mean  derived  from  the  observations 
of  several  years.  Thus,  at  Philadelphia,  Pa.,  the  mean 
annual  rainfall  is  about  40  inches,  but  in  1890  it  was  50.8 
inches  and  in  1885  it  was  only  33.4  inches.  Similarly  at 
Denver,  Col.,  the  mean  is  about  14  inches,  but  the  extremes 
are  about  20  and  9  inches.  When  a  very  low  annual  rain- 


348  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

fall  occurs,  that  of  the  year  preceding  or  following  is  also  apt 
to  be  low,  and  estimates  for  the  water  supply  of  towns  must 
take  into  account  this  minimum  annual  rainfall.  The  dis- 
tribution of  rainfall  throughout  the  year  must  also  be  con- 
sidered, and  for  this  purpose  the  rainfall  records  of  the  given 
locality  should  be  obtained  from  the  publications  of  the 
U.  S.  Weather  Bureau  and  be  carefully  discussed.  In 
making  plans  for  a  water  supply  it  should  be  the  aim  to 
secure  an  ample  amount  during  the  driest  months  of  the 
driest  year.  The  following  shows  the  average  rainfall  in 
inches  in  several  states  and  in  the  United  States  for  the 
four  seasons  and  for  the  year;  for  very  wet  years  about 
25  percent  may  be  added  to  these  values,  while  for  very 
dry  years  about  25  percent  may  be  subtracted. 


Spring. 

Summer. 

Autumn. 

Winter. 

Annual. 

Massachusetts, 

u.6 

11.4 

ii.  9 

II.7 

46.6 

Pennsylvania, 

10.3 

12.7 

10.0 

9-5 

42.5 

South  Carolina, 

9.8 

16.2 

9-7 

9-7 

454 

Illinois, 

TO.2 

II.  2 

9.0 

7-7 

38.1 

Minnesota, 

6-5 

10.8 

5-8 

3-1 

26.2 

Colorado, 

4.2 

5-5 

2.8 

2-3 

14.8 

California, 

6.2 

0-3 

3-5 

11.9 

21.9 

United  States, 

9.2 

10.3 

8-3 

8.6 

36.3 

After  rain  has  fallen  evaporation  immediately  begins 
from  both  land  and  water  surfaces,  and  this  continues  until 
all  the  rainfall  is  ultimately  evaporated  into  the  atmosphere, 
where  it  is  condensed  into  clouds  and  falls  again  as  rain. 
For  any  particular  watershed,  however,  the  evaporation  is 
less  than  the  rainfall,  so  that  the  remainder,  which  is  called 
" runoff,"  may  be  impounded  for  the  purposes  of  water 
supply  or  water  power.  In  the  Atlantic  states  the  annual 
evaporation  from  land  surfaces  is  about  40  percent  and  that 
from  water  surfaces  about  60  percent  of  the  annual  rainfall, 
so  that  about  one-half  of  the  annual  rainfall  may  be  utilized. 
In  the  arid  regions  west  of  the  Rocky  mountains  the  per- 
centages are  much  higher,  that  from  water  surfaces  being 
four  or  five  times  as  great  as  the  rainfall.  From  a  discus- 


ART.  135  GROUND    WATER    AND    RUNOFF  349 

sion  of  the  observed  evaporation  in  the  eastern  and  middle 
parts  of  the  United  States,  Vermeule  has  deduced  the  formula 

£  =  (15.  5+0.16^X0.057-  i.  48) 

where  R  is  the  annual  rainfall  and  E  the  annual  evaporation 
in  inches,  and  T  is  the  mean  annual  temperattire  in  Fahren- 
heit degrees.*  If  T  =  49°.  6,  this  becomes  £=15.5  +  0.16^, 
which  is  a  mean  value  for  New  Jersey  and  neighboring  states  ; 
if  T  be  47°  the  evaporation  is  ten  percent  less,  and  if  T  be 
52°  it  is  ten  percent  more,  than  this  mean. 

The  evaporation  in  different  months  varies  greatly,  the 
mean  monthly  temperature  being  the  controlling  factor. 
The  following  are  average  values  given  by  Vermeule  for  the 
vicinity  of  New  Jersey,  where  the  mean  annual  temperature 
is  49°.  6;  r  representing  mean  monthly  rainfall  and  e  mean 
monthly  evaporations  in  inches  : 

Jan.,  e=o.2j  +  o.ior  July,  e  =  3.oo+o.3or 

Feb.,  e=o.30+o.ior  Aug.,  e  =  2.62  +  o.2$r 

March,  e=o.48  +  o.ior  Sept.,  e  =  i.63+o.2or 

April,  e=o.87  +  o.ior  Oct.,  e=o.88  +  o.i2r 

May,  0  =  1.874-  o.2or  Nov.,  e=o.66+o.ior 

June,  e  =  2.5o+o.25r  Dec.,  e= 


To  obtain  the  monthly  evaporations  for  places  of  mean 
annual  temperature  T,  the  values  found  for  e  are  to  be  multi- 
plied by  0.057—1.48.  Thus,  if  there  be  8  inches  of  rain 
in  July,  e  =  5.40  inches,  and  if  the  mean  annual  temperature 
be  56°,  this  is  to  be  increased  by  32  percent. 

Prob.  134.  Show  that  one  inch  of  rainfall  per  month  is,  very 
nearly,  0.9  cubic  feet  per  second  per  square  mile. 

ART.  135.     GROUND  WATER  AND  RUNOFF 

When  the  ground  is  frozen  and  the  precipitation  does 
not  accumulate  in  the  form  of  ice  and  snow,  the  runoff  from 
a  watershed  is  closely  equal  to  the  rainfall  minus  the  evapo- 

*  U.  S.  Geological  Survey  of  New  Jersey  (Trenton,  1894),  vol.  3,  p.  76. 


350  WATER  SUPPLY  AND  WATER  POWER         CHAP,  xi 

ration.  If  three  inches  of  rain  falls  per  month  and  one- 
third  of  this  evaporates,  the  runoff  will  be  nearly  2  cubic 
feet  per  second  for  each  square  mile  of  the  watershed.  The 
discharge  due  to  a  heavy  rainfall  occurring  in  a  short  period 
or  to  the  melting  of  snow  may  be  twenty  or  thirty  times  as 
great.  A  rainfall  of  10  inches  occurring  in  two  days,  if  three- 
fourths  of  it  is  delivered  at  once  to  the  streams,  will  give  a 
flood  discharge  of  about  100  cubic  feet  per  second  per  square 
mile  of  watershed  area.  It  is  not  usually  necessary  to  con- 
sider these  flood  discharges  in  estimates  for  water  supply 
and  water  power,  except  in  order  to  take  precautions  against 
the  damage  they  may  cause. 

During  the  spring  the  ground  is  filled  with  water  which 
is  slowly  flowing  toward  the  streams,  and  this  ground  water 
is  the  main  source  of  the  runoff  from  a  watershed  during 
the  dry  months.  The  velocity  of  flow  of  this  ground  water 
varies  directly  as  the  slope  of  its  surface,  for  this  velocity 
is  slow  so  that  no  losses  occur  in  impact  (Art.  117).  When 
the  slope  of  the  surface  of  the  ground  water  becomes  zero, 
the  streams  are  dry  if  there  be  no  rainfall.  The  discharge 
of  a  stream  in  a  dry  season  hence  depends  upon  the  depth 
and  slope  of  the  ground  water,  and  this  in  turn  depends 
upon  the  previous  rainfall,  the  topography  of  the  country, 
and  the  character  of  the  soil. 

While  data  regarding  rainfall  and  evaporation  will  fur- 
nish valuable  information  regarding  the  mean  annual  flow 
of  a  stream,  they  will  usually  fail  to  indicate  the  mean  dis- 
charge during  different  months.  For  this  purpose  the 
study  of  discharge  curves  (Art.  126)  is  important,  and  if 
there  be  none  for  the  stream  in  hand,  it  will  be  necessary 
to  make  a  few  gagings  at  different  stages  of  water  and  to 
collect  information  regarding  the  lowest  stages  that  have 
been  observed  in  dry  years. 

In  irrigation  work  quantities  of  water  are  often  esti- 
mated in  terms  of  a  convenient  unit  called  the  '  *  acre-foot. ' ' 


ART.  135  GROUND    WATER   AND    RlJNOFF  351 

One  acre-foot  of  water  is  the  quantity  which  will  cover  one 
acre  to  a  depth  of  one  foot,  namely,  43  560  cubic  feet.  The 
discharge  of  a  stream  is  often  stated  in  acre-feet  per  day. 
One  acre-foot  per  day  is  0.5042  cubic  feet  per  second,  or 
one  cubic  foot  per  second  is  1.983  acre-feet  per  day.  One 
acre-foot  of  water  is  325851  U.  S.  gallons,  and  i  ooo  ooo 
gallons  is  3.0689  acre-feet. 

The  hydraulics  of  irrigation  engineering  differs  in  no  re- 
spect from  that  of  water  supply  and  water  power.  Water 
is  collected  in  reservoirs  or  obtained  by  damming  a  river,  and 
it  is  led  by  a  main  canal  to  the  area  to  be  irrigated,  and 
there  it  is  distributed  through  smaller  lateral  canals  to  the 
fields.  The  smaller  the  canal  or  ditch  the  steeper  becomes 
its  slope,  and  in  the  final  application  to  the  crops  the  flow 
in  the  furrows  is  often  normal  to  the  contours  of  the  sur- 
face. In  a  river  system  the  brooks  feed  the  creeks,  and 
the  creeks  feed  the  river,  the  flow  being  from  the  smaller 
to  the  larger;  in  an  artificial  irrigation  system,  however, 
the  flow  is  from  the  larger  to  the  smaller  channel. 

vSeepage  into  the  earth  from  an  irrigation  canal  con- 
stantly goes  on,  unless  its  bed  be  puddled  or  lined  with 
concrete,  and  this  loss  of  water  is  often  very  heavy.  For 
new  canals  it  is  often  as  high  as  50  percent  of  the  water, 
but  for  old  canals  it  may  become  lower  than  10  percent. 
In  making  estimates  for  an  irrigation-  supply  it  is  hence 
necessary  to  take  into  account  this  seepage  loss,  and  also 
to  consider  that  due  to  evaporation. 

In  irrigation  estimates  the  "duty"  of  water  is  to  be 
regarded.  This  is  defined  as  the  number  of  acres  that  can 
be  irrigated  by  a  supply  of  one  cubic  foot  per  second,  and 
it  usually  ranges  from  60  to  100  acres.  An  inverse  meas- 
ure of  duty  is  the  number  of  vertical  inches  of  water  required 
to  irrigate  any  area,  this  usually  ranging  from  1 8  to  24  inches. 
The  acre-foot  is  also  frequently  used  in  statements  of  duty, 
one  acre-foot  per  acre  being  the  same  as  12  vertical  inches 


352  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

of  water.  The  methods  of  measuring  the  water  by  orifices 
and  modules  in  terms  of  the  miner's  inch  unit  have  been 
explained  in  Art.  50. 

Prob.  135.  If  all  the  rainfall  that  does  not  evaporate  flows 
into  the  streams,  find  the  runoff  in  cubic  feet  per  second  from 
a  watershed  of  1225  square  miles  during  a  month  when  the 
rainfall  is  3.6  inches,  the  mean  annual  temperature  being  48°. 5 
Pahrenheit.  Also  for  the  temperature  of  49°.  5. 

ART.  136.      ESTIMATES  FOR  WATER  SUPPLY 

The  consumption  of  water  in  American  cities  is,  on  the 
average,  about  100  gallons  per  person  per  day,  the  large 
cities  using  more  and  the  small  ones  less  than  this  amount. 
The  daily  consumption  in  July  and  August  is  from  15  to 
20  percent  greater  than  the  mean,  owing  to  the  use  of  water 
for  sprinkling,  while  during  January  and  February  it  is  also 
greater  than  the  mean  in  the  colder  localities  owing  to  the 
large  amount  that  is  allowed  to  run  to  waste  in  houses  in 
order  to  prevent  the  freezing  of  the  pipes  On  Mondays, 
when  every  household  is  at  work  on  the  weekly  washing, 
the  consumption  may  be  put  at  50  percent  higher  than  the 
mean  for  the  week.  Accordingly  if  the  yearly  mean  be 
100  gallons  per  person  per  day,  the  Monday  consumption 
during  very  hot  or  very  cold  weather  may  be  as  high  as  150 
gallons  per  person  per  day.  When  a  large  fire  occurs  the 
hourly  consumption  for  this  purpose  alone  in  a  fire  district 
of  i  o  ooo  people  may  be  at  the  rate  of  1 7  5  gallons  per  per- 
son per  day.  In  general  the  maximum  available  hourly 
supply  should  be  from  three  to  four  times  as  great  as  that 
of  the  mean  daily  consumption. 

When  water  is  to  be  pumped  from  a  river  directly  into 
the  pipes,  without  tank  or  reservoir  storage,  the  capacity 
of  the  pumps  should  be  such  that  during  the  occurrence 
of  fires  at  least  three  times  the  mean  daily  consumption 


ART.  136  ESTIMATES   FOR   WATER  SUPPLY  353 

may  be  furnished.  When  a  pump  delivers  water  to  a  dis- 
tributing reservoir,  its  capacity  need  not  be  so  high  as  in 
the  case  of  direct  pumping,  for  the  reservoir  storage  can  be 
drawn  upon  in  case  of  fire.  When  the  reservoir  is  large  the 
pump  capacity  need  be  only  sufficient  to  lift  the  annual 
consumption  during  the  time  when  it  is  in  operation.  The 
subject  of  pumping  is  an  extensive  one,  but  it  will  be  briefly 
treated  from  a  hydraulic  standpoint  in  Arts.  187  and  191. 

Gravity  supplies  are  those  furnished  by  impounding  the 
runoff  of  a  watershed  in  a  reservoir.  As  an  example  of  a 
preliminary  investigation  of  such  a  case,  suppose  a  town 
of  6000  people  desires  to  obtain  a  mean  supply  of  100  gal- 
lons per  person  per  day  or  in  total  1.86  acre-feet  per  day 
(Art.  135).  A  certain  watershed  a  few  miles  from  the 
town  has  an  area  of  1410  acres,  and  the  minimum  annual 
rainfall  is  31  inches,  of  which  15  inches  evaporates.  The 
available  storage  is  hence  1880  acre-feet,  or  5.1  acre-feet 
per  day,  and  accordingly  sufficient  water  can  be  obtained 
for  the  supply  of  the  town  if  storage  capacity  can  be  pro- 
vided. To  estimate  the  capacity  of  the  reservoir,  suppose 
that  in  the  dry  years  August  is  a  wet  month,  so  that  the 
reservoir  may  be  full  at  the  end  of  that  month.  Let  the 
September  rainfall  be  1.2  inches,  of  which  40  percent  evap- 
orates, and  the  October  rainfall  be  0.6  inches,  of  which  30 
percent  evaporates.  Then  during  September  the  maxi- 
mum available  storage  is  84.6  acre-feet,  while  the  consump- 
tion is  55.8  acre-feet,  so  that  the  reservoir  is  also  full  at 
the  end  of  that  month.  During  October,  however,  the 
available  storage  is  49.3  acreTfeet,  while  the  consumption 
is  57.7  acre-feet,  and  hence  the  deficiency  of  8.4  acre-feet 
must  be  provided  for  by  storing  the  September  rainfall. 
The  capacity  of  the  reservoir,  therefore,  must  be  more  than 
8.4  acre-feet;  if  it  is  to  be  half -full  at  the  end  of  October, 
its  capacity  must  be  16.8  acre-feet,  or  about  5  400  ooo 
gallons. 


354  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

The  above  estimate,  being  based  on  rainfall  and  evapora- 
tion records,  is  liable  to  some  uncertainty,  for  it  has  been 
assumed  that  all  the  rainfall  that  does  not  evaporate  reaches 
the  reservoir.  This  uncertainty  can  be  removed  by  gaging 
the  stream  at  the  place  where  it  is  proposed  to  build  the 
reservoir.  Three  gagings  made  at  high,  medium,  and  very 
low  stages  of  water  will  furnish  data  from  which  a  discharge 
curve  can  be  drawn  (Art.  126),  and  from  this,  in  connection 
with  records  of  the  stages  in  dry  months  of  dry  years,  a 
much  more  reliable  estimate  of  reservoir  capacity  may  be 
made.  For  instance,  suppose  the  September  discharge 
is  found  to  be  i.io  cubic  feet  per  second,  or  65.4  acre-feet, 
and  the  October  discharge  0.75  cubic  feet  per  second,  or 
46.1  acre-feet;  then  it  is  seen  that  the  reservoir  capacity 
should  be  about  40  percent  greater  than  the  previous  esti- 
mate. After  the  height  of  the  water  level  of  the  reservoir 
is  fixed  the  dimensions  of  its  waste-weir  may  be  computed 
from  Art.  68,  and  the  size  of  the  main  pipe  line  by  Art.  93 ; 
for  the  latter  computation  proper  pressures  must  be  assumed 
throughout  the  town,  so  that  ample  head  may  be  provided 
for  fire  contingencies.  When  the  main  divides  into  branches 
the  problem  of  computing  the  diameters  from  the  given 
data  is  indeterminate  (Art.  100),  and  hence  it  will  probably 
be  as  well  to  assume  at  the  outset  the  sizes  of  the  main  and 
its  branches.  The  velocities  corresponding  to  the  given 
quantities  and  the  assumed  sizes  are  then  computed,  and 
from  these  the  pressure-heads  at  a  number  of  points  are 
found.  If  these  are  not  satisfactory,  other  sizes  are  to  be 
taken  and  the  computation  be  repeated.  The  successful 
design  will  be  that  which  will  furnish  the  required  quantities 
under  proper  pressures  with  the  least  expenditure. 

Prob.  136.  Let  the  main  for  the  above  case  be  4  inches  in 
diameter  and  15  ooo  feet  long,  and  the  fall  be  175  feet.  Com- 
pute the  pressure-head  in  the  town  when  the  consumption  is 
150  gallons  per  person  per  day. 


ART.  137  ESTIMATES    FOR    WATER    POWER  355 


ART.  137.     ESTIMATES  FOR  WATER  POWER 

The  methods  of  estimating  the  water  power  that  can  be 
derived  by  damming  a  stream  are  similar  to  those  for  water 
supply.  In  the  absence  of  gagings  the  records  of  rainfall 
and  evaporation  are  to  be  collected  and  discussed,  but  a 
few  gagings  will  give  much  more  definite  information  if 
records  of  water  stages  during  several  years  can  be  had. 
Here  also  the  minimum  flow  of  the  stream  must  receive 
careful  attention,  particularly  when  the  plant  is  to  generate 
electric  power  for  trolley  and  light  service,  for  the  inter- 
ruption of  such  service  is  a  serious  public  inconvenience. 
It  has  frequently  happened,  indeed,  that  a  water-power 
plant  built  without  sufficient  investigation  has  proved 
unable  to  furnish  sufficient  power  during  dry  seasons,  and 
it  has  been  necessary  to  install  an  auxiliary  steam  plant 
to  make  good  the  deficiency. 

Let  W  be  the  weight  of  water  delivered  per  second  to  a 
hydraulic  motor,  and  h  be  its  effective  head  as  it  enters  the 
motor,  h  being  due  either  to  pressure  (Art.  11),  or  to  veloc- 
ity (Art.  22),  or  to  pressure  and  velocity  combined  (Art.  25). 
The  theoretic  energy  per  second  of  this  water  is 

K=Wh  (137), 

and  if  W  be  in  pounds  and  k  in  feet,  the  theoretic  horse- 
power of  the  water  as  it  enters  the  motor  is 

(137), 


and  this  is  the  power  that  can  be  developed  by  a  motor  of 
efficiency  unity.  The  work  k  delivered  by  the  motor  is, 
however,  always  less  than  K  owing  to  losses  in  impact  and 
friction,  and  the  horse-power  hp  of  the  motor  is  less  than 
HP.  The  efficiency  of  the  motor  is 

e=k/K=k/Wh        or        e  =hp/HP          (137), 
and  the  value  of  this  for  turbine  wheels  is  usually  about 


356  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

0.75,  that  is,  the  wheel  transforms  into  useful  work  about 
75  percent  of  the  energy  of  the  water  that  enters  it. 

In  designing  a  water-power  plant  it  should  be  the  aim 
to  arrange  the  forebays  and  penstocks  which  lead  the  water 
to  the  wheel  so  that  the  losses  in  these  approaches  may 
be  as  small  as  possible.  The  entrance  from  the  head  race 
into  the  forebay,  from  the  forebay  into  the  penstock,  and 
from  the  penstock  to  the  motor,  should  be  smooth  and  well 
rounded ;  sudden  changes  in  cross-section  should  be  a  voided , 
and  all  velocities  should  be  low  except  that  at  the  motor. 
If  these  precautions  be  carefully  observed,  the  loss  of  head 
outside  of  the  motor  can  be  made  very  small.  Let  H  be 
the  total  head  from  the  water  level  in  the  head  race  to  that 
in  the  tail  race  below  the  motor.  The  total  available  en- 
ergy per  second  is  WH,  and  it  should  be  the  aim  of  the 
designer  to  render  the  losses  of  head  in  the  approaches  as. 
small  as  possible  so  that  the  effective  head  h  may  be  as 
nearly  equal  to  H  as  possible.  Neglect  of  these  precautions 
may  render  the  effective  power  less  than  that  estimated. 

The  efficiency  ^  of  the  approaches  is  the  ratio  of  the 
energy  K  of  the  water  as  it  enters  the  wheel  to  the  maxi- 
mum available  energy  WH,  or  ^  =K/WH.  The  efficiency 
.E  of  the  entire  plant,  consisting  of  both  approaches  and 
wheel,  is  the  ratio  of  the  work  k  delivered  by  the  wheel  to 
the  energy  WH,  or 

E  =k/WH  =eK/WH  =ee^ 

or,  the  final  efficiency  is  the  product  of  the  separate  effi- 
ciencies. If  the  efficiency  of  the  wheel  be  0.75  and  that 
of  the  approaches  0.96,  the  efficiency  of  the  plant  as  a 
whole  is  0.72,  or  only  72  percent  of  the  theoretic  energy  is 
utilized.  Usually  the  efficiency  of  the  approaches  can  be 
made  higher  than  96  percent. 

In  making  estimates  for  a  proposed  plant,  the  efficiency 
of  turbine  wheels  may  be  taken  at  75  percent ;  the  effective 


ART.  138  WATER   DELIVERED    TO   A   MOTOR  357 

work  is  then  0.75!^,  and  accordingly  if  the  wheels  are 
required  to  deliver  the  work  k  per  second,  the  approaches 
are  to  be  so  arranged  that  Wh  shall  not  be  less  than  1.33^. 
Especially  when  the  water  supply  is  limited  it  is  important 
to  make  all  efficiencies  as  high  as  possible. 

Prob.  137.  A  stream  delivers  500  cubic  feet  of  water  per 
second  to  a  canal  which  terminates  in  a  forebay  where  the 
water  level  is  8.1  feet  above  the  tail  race.  The  wheels  deliver 
335  horse-powers  and  their  efficiency  is  known  to  be  75  percent. 
How  much  power  is  lost  in  the  forebay  and  penstock  ? 

ART.  138.     WATER  DELIVERED  TO  A  MOTOR 

To  determine  the  efficiency  of  a  hydraulic  motor  by  for- 
mula (137)  3,  k  is  to  be  measured  by  the  methods  of  Art.  140, 
and  h  found  by  Art.  139.  In  order  to  find  the  weight  W 
that  passes  through  the  wheel  in  one  second,  there  must 
be  known  the  discharge  per  second  g-and  the  weight  w  of  a 
cubic  unit  of  water;  then 

W =wq 

Here  w  may  be  found  by  weighing  one  cubic  foot  of  the 
water,  or  when  the  water  contains  few  impurities  its  tem- 
perature may  be  noted  and  the  weight  be  taken  from 
Table  7.  In  approximate  computations  w  may  be  taken 
at  62.5  pounds  per  cubic  foot.  In  precise  tests  of  motors, 
however,  its  actual  value  should  be  ascertained  as  closely 
as  possible. 

The  measurement  of  the  flow  of  water  through  orifices, 
weirs,  tubes,  pipes,  and  channels  has  been  so  fully  discussed 
in  the  preceding  chapters,  that  it  only  remains  here  to 
mention  one  or  two  simple  methods  applicable  to  small 
quantities,  and  to  make  a  few  remarks  regarding  the  sub- 
ject of  leakage.  In  any  particular  case  that  method  of 
determining  q  is  to  be  selected  which  will  furnish  the  re- 
quired degree  of  precision  with  the  least  expense  (Art.  125). 


358  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

For  a  small  discharge  the  water  may  be  allowed  to  fall 
into  a  tank  of  known  capacity.  The  tank  should  be  of 
uniform  horizontal  cross-section,  whose  area  can  be  accu- 
rately determined,  and  then  the  heights  alone  need  be 
observed  in  order  to  find  the  volume.  These  in  precise 
work  will  be  read  by  hook  gages,  and  in  cases  of  less  accu- 
racy by  measurements  with  a  graduated  rod.  At  the  be- 
ginning of  the  experiment  a  sufficient  quantity  of  water 
must  be  in  the  tank  so  that  a  reading  of  the  gage  can  be 
taken;  the  water  is  then  allowed  to  flow  in,  the  time 
between  the  beginning  and  end  of  the  experiment  being 
determined  by  a  stop-watch,  duly  tested  and  rated.  This 
time  must  not  be  short,  in  order  that  the  slight  errors  in 
reading  the  watch  may  not  affect  the  result.  The  gage  is 
read  at  the  close  of  the  test  after  the  surface  of  the  water 
becomes  quiet,  and  the  difference  of  the  gage-readings  gives 
the  depth  which  has  flowed  in  during  the  observed  time. 
The  depth  multiplied  by  the  area  of  the  cross-section  gives 
the  volume,  and  this  divided  by  the  number  of  seconds 
during  which  the  flow  occurred  furnishes  the  discharge  per 
second  q. 

If  the  discharge  be  very  small,  it  may  be  advisable  to 
weigh  the  water  rather  than  to  measure  the  depths  and 
cross-sections.  The  total  weight  divided  by  the  time  of 
flow  then  gives  directly  the  weight  W.  This  has  the  advan- 
tage of  requiring  no  temperature  observation,  and  is  proba- 
bly the  most  accurate  of  all  methods,  but  unfortunately 
it  is  not  'possible  to  weigh  a  considerable  volume  of  water 
except  at  great  expense. 

When  water  is  furnished  to  a  motor  through  a  small  pipe 
a  common  water  meter  may  often  be  advantageously  used 
to  determine  the  discharge  (Art.  38).  No  water  meter, 
however,  can  be  regarded  as  accurate  until  it  has  been 
tested  by  comparing  the  discharge  as  recorded  by  it  with 
the  actual  discharge  as  determined  by  measurement  or 


ART.  138  WATER    DELIVERED    TO    A    MOTOR  359 

weighing  in  a  tank.  Such  a  test  furnishes  the  constants  for 
correcting  the  result  found  by  its  readings,  which  otherwise 
is  liable  to  be  5  or  10  percent  in  error. 

The  leakage  which  occurs  in  the  flume  or  penstock  before 
the  water  reaches  the  wheel  should  not  be  included  in  the 
value  of  W,  which  is  used  in  computing  its  efficiency, 
although  it  is  needed  in  order 'to  ascertain  the  efficiency  of 
the  entire  plant.  The  manner  of  determining  the  amount 
of  leakage  will  vary  with  the  particular  circumstances  of 
the  case  in  hand.  If  it  be  very  small,  it  may  be  caught  in 
pails  and  directly  weighed.  If  large  in  quantity,  the  gates 
which  admit  water  to  the  wheel  may  be  closed,  and  the 
leakage  being  then  led  into  the  tail  race  it  may  be  there 
measured  by  a  weir,  or  by  allowing  it  to  collect  in  a  tank. 
The  leakage  from  a  vertical  penstock  whose  cross-section 
is  known  may  be  ascertained  by  filling  it  with  water,  the 
wheel  being  still,  and  then  observing  the  fall  of  the  water 
level  at  regular  intervals  of  time.  In  designing  construc- 
tions to  bring  water  to  a  motor,  it  is  best,  of  course,  to 
arrange  them  so  that  all  leakage  will  be  avoided,  but  this 
cannot  often  be  fully  attained,  except  at  great  expense. 

The  most  common  method  of  measuring  q  is  by  means  of 
a  weir  placed  in  the  tail  race  below  the  wheel.  This  has 
the  disadvantage  that  it  sometimes  lessens  the  fall  which 
would  be  otherwise  available,  and  that  often  the  velocity 
of  approach  is  high.  It  has,  however,  the  advantage  of 
cheapness  in  construction  and  operation,  and  for  any  con- 
siderable discharge  appears  to  be  almost  the  only  method 
which  is  both  economical  and  precise.  If  the  weir  is 
placed  above  the  wheel,  the  leakage  of  the  penstock  must 
be  carefully  ascertained. 

Prob.  138.  A  weir  with  end  contractions  and  no  velocity  of 
approach  has  a  length  of  1.33  feet,  and  the  depth  on  the  crest 
is  0.406  feet.  The  same  water  passes  through  a  small  turbine 


360  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

under  the  effective  head  10.49  feet-  Compute  the  theoretic 
horse-power. 

ART.  139.     EFFECTIVE  HEAD  ON  A  MOTOR 

The  total  available  head  H  between  the  surface  of  the 
water  in  the  reservoir  or  head  race  and  that  in  the  lower 
pool  or  tail  race  is  determined  by  running  a  line  of  levels 
from  one  to  the  other.  Permanent  bench  marks  being 
established,  gages  can  then  be  set  in  the  head  and  tail  races 
and  graduated  so  that  their  zero  points  will  be  at  some 
datum  below  the  tail-race  level.  During  the  test  of  a 
wheel  each  gage  is  read  by  an  observer  at  stated  intervals, 
and  the  difference  of  tHe  readings  gives  the  head  H .  In 
some  cases  it  is  possible  to  have  a  floating  gage  on  the  lower 
level,  the  graduated  rod  of  which  is  placed  alongside  a 
glass  tube  that  communicates  with  the  upper  level;  the 
head  H  is  then  directly  read  by  noting  the  point  of  the 
graduation  which  coincides  with  the  water  surface  in  the 
tube.  This  device  requires  but  one  observer,  while  the 
former  requires  two;  but  it  is  usually  not  the  cheapest 
arrangement  unless  a  large  number  of  observations  are  to 
be  taken. 

From  this  total  head  H  are  to  be  subtracted  the  losses 
of  head  in  entering  the  forebay  and  penstock,  and  the  loss 
of  head  in  friction  in  the  penstock  itself,  and  these  losses 
may  be  ascertained  by  the  methods  of  Chapters  VIII  and 

IX.     Then 

fc-H -*'--*" 

is  the  effective  head  acting  upon  the  wheel.  In  properly 
designed  approaches  the  lost  heads  hf  and  h"  are  very 
small. 

When  water  enters  upon  a  wheel  through  an  orifice 
which  is  controlled  by  a  gate,  losses  of  head  will  result, 
which  can  be  estimated  by  the  rules  of  Chapters  V  and  VI. 
If  this  orifice  is  in  the  head  race,  the  loss  of  head  should  be 


ART.  139  EFFECTIVE    HEAD    ON   A   MOTOR  361 

subtracted  together  with  the  other  losses  from  the  total 
head  H.  But  if  the  regulating  gates  are  a  part  of  the  wheel 
itself,  as  is  the  case  in  a  turbine,  the  loss  of  head  should  not 
be  subtracted,  because  it  is  properly  chargeable  to  the  con- 
struction of  the  wheel,  and  not  to  the  arrangements  which 
furnish  the  supply  of  water.  In  any  event  that  head 
should  be  determined  which  is  to  be  used  in  the  subsequent 
discussions  :  if  the  efficiency  of  the  fall  is  desired,  the  total 
available  head  is  required;  if  the  efficiency  of  the  motor, 
that  effective  head  is  to  be  found  which  acts  directly  upon 
it  (Art.  139). 

When  water  is  delivered  through  a  nozzle  or  pipe  to  an 
impulse  wheel,  the  head  h  is  not  the  total  fall,  since  a  large 
part  of  this  may  be  lost  in  friction  in  the  pipe,  but  is  merely 
the  velocity-head  v*/2g  of  the  issuing  jet.  The  value  of  v 
is  known  when  the  discharge  q  and  the  area  of  the  cross- 
section  of  the  stream  have  been  determined,  and 


2g 

In  the  same  manner  when  a  stream  flows  in  a  channel  against 
the  vanes  of  an  undershot  wheel  the  effective  head  is  the 
velocity-head,  and  the  theoretic  energy  is 


_ 

2g     2ga 

If,  however,  the  water  in  passing  through  the  wheel  falls  a 
distance  h0  below  the  mouth  of  the  nozzle,  then  the  effective 
head  which  acts  upon  the  wheel  is  given  by 


In  order  to  fully  utilize  the  fall  h0  it  is  plain  that  the  wheel 
should  be  placed  as  near  the  level  of  the  tail  race  as  possible. 

Lastly,  when  water  enters  a  turbine  wheel  through  a 
pipe,  a  piezometer  may  be  placed  near  the  wheel  entrance 
to  register  the  pressure-head  during  the  flow  ;  if  this  pres- 


362  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

sure-head,  measured  upon  and  from  the  water  level  in  the 
tail  race,  be  called  ht  and  if  the  velocity  in  the  pipe  be  v,  then 


is  the  effective  head  acting  on  the  wheel.  It  is  here  sup- 
posed that  the  turbine  has  a  draft  tube  leading  below  the 
water  level  in  the  tail  race  ;  if  this  is  not  the  case,  ht  should 
be  measured  upward  from  the  lowest  part  oi  the  exit  orifices. 

Prob.  139.  A  pressure  gage  at  the  entrance  of  a  nozzle  regis- 
ters 116  pounds  per  square  inch,  and  the  coefficient  of  velocity 
of  the  nozzle  is  0.98.  Compute  the  effective  velocity-head  of 
the  issuing  jet. 

ART.  140.     MEASUREMENT  OF  EFFECTIVE  POWER 

The  effective  work  and  horse-power  delivered  by  a 
water-wheel  ,or  hydraulic  motor  is  often  required  to  be 
measured.  Water-power  may  be  sold  by  means  of  the 
weight  W,  or  quantity  g,  furnished  under  a  certain  head, 
leaving  the  consumer  to  provide  his  own  motor  ;  or  it  may 
be  sold  directly  by  the  number  of  horse-power.  In  either 
case  tests  must  be  made  from  time  to  time  in  order  to  in- 
sure that  the  quantity  contracted  for  is  actually  delivered 
and  is  not  exceeded.  It  is  also  frequently  required  to  meas- 
ure effective  work  in  order  to  ascertain  the  power  and 
efficiency  of  the  motor,  either  because  the  party  who  buys 
it  has  bargained  for  a  certain  power  and  efficiency,  or  because 
it  is  desirable  to  know  exactly  what  the  motor  is  doing  in 
order  to  improve  if  possible  its  performance. 

The  test  of  a  hydraulic  motor  has  for  its  object:  first, 
the  determination  of  the  effective  energy  and  power; 
second,  the  determination  of  its  efficiency;  and  third, 
the  determination  of  that  speed  which  gives  the  greatest 
power  and  efficiency.  If  the  wheel  be  still,  there  is  no 
power;  if  it  be  revolving  very  fast,  the  water  is  flowing 
through  it  so  as  to  change  but  little  of  its  energy  into  work  : 


ART.  140         MEASUREMENT  OF  EFFECTIVE  POWER 


363 


and  in  all  cases  there  is  found  a  certain  speed  which  gives 
the  maximum  power  and  efficiency.  To  execute  these  tests, 
it  is  not  at  all  necessary  to  know  how  the  motor  is  con- 
structed or  the  principle  of  its  action,  although  such  knowl- 
edge is  very  valuable,  and  is  in  fact  indispensable,  in  order 
to  enable  the  engineer  to  suggest  methods  by  which  its 
operation  may  be  improved. 

A  method  in  which  the  effective  work  of  a  small  motor 
may  be  measured  is  to  compel  it  to  exert  all  its  power  in 
lifting  a  weight.  For  this  purpose  the  weight  may  be 
attached  to  a  cord  which  is  fastened  to  the  horizontal  axis 
of  the  motor,  and  around  which  it  winds  as  the  shaft  re- 
volves. The  wheel  then  expends  all  its  power  in  lifting; 
this  weight  Wt  through  the  height  hl  in  t{  seconds,  and 
the  work  performed  per  second  then  is  k  =Wlh1/tl.  This 
method  is  rarely  used  in  practice  on  account  of  the  diffi- 
culty of  measuring  ^  with  precision. 

The  usual  method  of  measuring  the  effective  work  of 
a  hydraulic  motor  is  by  means  of  the  friction  brake  or 
power  dynamometer  invented  by  Prony  about  1780.  In 
Fig.  140  is  illustrated  a  simple 
method  of  applying  the  appa- 
ratus to  a  vertical  shaft,  the 
upper  diagram  being  a  plan 
and  the  lower  an  elevation. 
Upon  the  vertical  shaft  is  a 
fixed  pulley,  and  against  this 
are  seen  two  rectangular  pieces 
of  wood  hollowed  so  as  to  fit 
it,  and  connected  by  two  bolts. 
By  turning  the  nuts  on  these 
bolts  while  the  pulley  is  re- 
volving, the  friction  can  be 
increased  at  pleasure,  even  so 
as  to  stop  the  motion ;  around 
these  bolts  between  the  blocks  are  two  spiral  springs  (not 


364  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

i 

shown  in  the  diagram)  which  press  the  blocks  outward  when 
the  nuts  are  loosened.  To  one  of  these  blocks  is  attached 
a  cord  which  runs  horizontally  to  a  small  movable  pulley 
over  which  it  passes,  and  supports  a  scale-pan  in  which 
weights  are  placed.  This  cord  runs  in  a  direction  opposite 
to  the  motion  of  the  shaft,  so  that  when  the  brake  is  tight- 
ened it  is  prevented  from  revolving  by  the  tension  caused 
by  the  weights.  The  direction  of  the  cord  in  the  horizontal 
plane  must  be  such  that  the  perpendicular  let  fall  upon  it 
from  the  center  of  the  shaft,  or  its  lever-arm,  is  constant; 
this  can  be  effected  by  keeping  the  small  pointer  on  the 
brake  at  a  fixed  mark  established  for  that  purpose. 

To  measure  the  work  done  by  the  wheel,  the  shaft  is  dis- 
connected from  the  machinery  which  it  usually  runs,  and 
allowed  to  revolve,  transforming  all  its  work  into  heat  by 
the  friction  between  the  revolving  pulley  and  the  brake, 
which  is  kept  stationary  by  tightening  the  nuts,  and  at 
the  same  time  placing  sufficient  weights  in  the  scale-pan  to 
hold  the  pointer  at  the  fixed  mark.  Let  n  be  the  number 
of  revolutions  per  second  as  determined  by  a  counter 
attached  to  the  shaft,  P  the  tension  in  the  cord  which  is 
equal  to  the  weight  of  the  scale-pan  and  its  loads,  /  the  lever- 
arm  of  this  tension  with  respect  to  the  center  of  the  shaft, 
r  the  radius  of  the  pulley,  and  F  the  total  force  of  friction 
between  the  pulley  and  the  brake.  Now  in  one  revolution 
the  force  F  is  overcome  through  the  distance  2nr,  and  in  n 
revolutions  through  the  distance  2xrn.  Hence  the  effec- 
tive work  done  by  the  wheel  in  one  second  is 

k  =jp.  2-nrn  = 


The  force  F  acting  with  the  lever-arm  r  is  exactly  balanced 
by  the  force  P  acting  with  the  lever-arm  /;  accordingly 
the  moments  Fr  and  PI  are  equal,  and  hence  the  work 
done  by  the  wheel  in  one  second  is 


ART.  140        MEASUREMENT  OF  EFFECTIVE  POWER  365 

If  P  be  in  pounds  and  /  in  feet,  the  effective  horse-power 
of  the  wheel  is  given  by 


As  the  number  of  revolutions  in  one  second  cannot  be  ac- 
curately read,  it  is  usual  to  record  the  counter  readings 
every  minute  or  half-minute  ;  if  N  be  the  number  of  revo- 
lutions per  minute, 

hp  =  27rATP//33  ooo  (140)2 

It  is  seen  that  this  method  is  independent  of  the  radius  of 
the  pulley,  which  may  be  of  any  convenient  size;  for  a 
small  motor  the  brake  may  be  clamped  directly  upon  the 
shaft,  but  for  a  large  one  a  pulley  of  considerable  size  is 
needed,  and  a  special  arrangement  of  levers  is  used  instead 
of  a  cord. 

The  efficiency  of  the  motor  is  now  found  by  dividing  the 
-effective  work  per  second  by  the  theoretic  work  per  second. 
Let  K  be  this  theoretic  work,  which,  is  expressed  by  Wh, 
where  W  and  h  are  determined  by  the  methods  of  Arts. 
138  and  139  ;  then 

e=k/K        or        e=Tip/HP 

The  work  measured  by  the  friction  brake  is  that  delivered 
at  the  circumference  of  the  pulley,  and  does  not  include 
that  power  which  is  required  to  overcome  the  friction  of 
the  shaft  upon  its  bearings.  The  shaft  or  axis  of  every 
water  wheel  must  have  at  least  two  bearings,  the  friction 
of  which  consumes  probably  about  2  or  3  percent  of  the 
power.  The  hydraulic  efficiency  of  the  wheel,  regarded  as 
a  user  of  water,  is  hence  2  or  3  percent  greater  than  the 
computed  value  of  e. 

There  are  in  use  various  forms  and  varieties  of  the  fric- 
tion brake,  but  they  all  act  upon  the  principle  and  in  the 
manner  above  described.  For  large  wheels  they  are  made 
of  iron,  and  almost  completely  encircle  the  pulley;  while 


366  WATER  SUPPLY  AND  WATER  POWER         CHAP,  xi 

a  special  arrangement  of  levers  is  used  to  lift  the  large 
weight  P.*  If  the  work  transformed  into  friction  be  large, 
both  the  brake  and  the  pulley  may  become  hot,  to  prevent 
which  a  stream  of  cool  water  is  allowed  to  flow  upon  them. 
To  insure  steadiness  of  motion,  it  is  well  that  the  surface 
of  the  pulley  should  be  lubricated,  which  for  a  wooden 
brake  is  well  done  by  the  use  of  soap.  It  is  important 
that  the  connection  of  the  cord  to  the  brake  should  be 
so  made  that  the  lever-arm  I  increases  when  the  brake 
moves  slightly  with  the  wheel ;  if  this  is  not  done,  the  wheel 
will  be  apt  to  cause  the  brake  to  revolve  with  it. 

Prob.  140a.  What  is  the  horse-power  of  a  motor  which  in 
75.5  seconds  lifts  a  weight  of  320  pounds  through  a  vertical 
height  of  42  feet? 

Prob.  1406.  Find  the  power  and  efficiency  of  a  motor  when 
the  theoretic  energy  is  1.38  horse-power,  which  makes  670  revo- 
lutions per  minute,  the  weight  on  the  brake  being  2  pounds  14. 
ounces  and  its  lever- arm  1.33  feet. 

ART.  141.     TESTS  OF  TURBINE  WHEELS 

The  following  description  of  a  test  of  a  6 -inch  Eureka 
turbine  may  serve  to  exemplify  the  methods  of  the  pre- 
ceding articles.  The  water  was  measured  by  a  weir  from 
which  it  ran  into  a  vertical  penstock  15.98  square  feet  in 
horizontal  cross-section.  This  plan  of  having  the  weir 
above  the  wheel  is  not  a  good  one,  but  it  was  here  adopted 
on  account  of  lack  of  room  below  the  turbine.  When  a 
constant  head  was  maintained  in  the  penstock  the  quantity 
of  water  flowing  through  the  wheel  was  the  same  as  that 
passing  the  weir;  if,  however,  the  head  in  the  penstock 
fell  oo  feet  per  minute  the  flow  through  the  wheel  in  cubic 
feet  per  minute  was  6og  +  15.98*,  in  wh;ch  q  is  the  discharge 
per  second  over  the  weir.  As  the  supply  of  water  was  very 

*  Thurston,  Transactions  American  Society  Mechanical  Engineers,  i886> 
vol.  8,  p.  359. 


ART.  141  TESTS    OF   TURBINE    WHEELS  367 

limited  the  wheel  could  not  be  run  to  its  full  capacity. 
The  level  of  water  in  the  penstock  was  read  upon  a  head 
gage  consisting  of  a  glass  tube  behind  which  a  graduated 
scale  was  fixed,  the  zero  of  which  was  a  little  above  the 
water  level  in  the  tail  race.  The  latter  level  was  read  upon  a 
fixed  graduated  scale  having  its  zero  in  the  same  horizontal 
plane  as  the  first;  these  readings  were  hence  essentially 
negative.  The  head  upon  the  wheel  is  then  found  by  add- 
ing the  readings  of  the  two  gages. 

The  vertical  shaft  of  the  turbine,  being  about  15  feet 
long,  was  supported  by  four  bearings,  and  to  a  small  pulley 
upon  its  upper  end  was  attached  the  friction  dynamometer 
as  described  in  the  last  article.  The  number  of  revolutions 
was  read  from  a  counter  placed  in  the  top  of  this  shaft. 
The  observations  were  taken  at  minute  intervals,  electric 
bells  giving  the  signals,  so  that  precisely  at  the  beginning 
of  each  minute  simultaneous  readings  were  taken  by  ob- 
servers at  the  weir,  at  the  head  gage,  at  the  tail  gage,  and 

Time  on      Depth     Penstock       Tail-race     *e™lu;       Wei«ht 

April  1 3,    °£r^ir      Gage.  Gage.        ^  in        Bf°n  Remarks. 

_QQQ  v^resb.  T?£n=»+-  17^^-4-  ^jnc  jjraKC. 

>88'          Feet.  eet-        Minute.      Pounds. 

7h     I7m       0.288  II.2<5  —0.21  2.5 

615  Length  of  weir, 

18  0.289       11.17  0.20  2.5  .' 

625  0  =  1. 909  feet. 

19  0.289       11.13          0.21  2.<  T      __.     ,, 

675  Length  of  lever- 

20  0.288       ii. 10          0.21  2.1; 

arm  on  brake, 

3h   22m  0.287  10.81  —0.20  3.0  /=i.43i  feet. 

23  0.287  10.69  0.20  5«           3.o  Gate   of    wheel   f 

24  0.287  10.62  0.21  3.0  open  during  all  ex- 

25  0.286  10.57  0.21  3.0  Periments. 

at  the  counter,  the  operator  at  the  brake  continually  keep- 
ing it  in  equilibrium  with  the  resisting  weight  in  the  scale- 
pan  by  slightly  tightening  and  loosening  the  nuts  as  re- 
quired. The  -above  shows  notes  of  all  the  observations 
of  two  sets  of  tests,  each  lasting  three  minutes,  the  weight 
in  the  scale-pan  being  different  in  the  two  sets. 

The  following  are  the  results  of  the  computations  made 
from  the  above  notes  for  each  minute  interval.  The  second 


368  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

column  is  derived  from  formula  (63)^  using  the  coefficient 
corresponding  to  the  given  length  of  weir  and  depth  on 
crest.  The  third  column  is  obtained  by  taking  the  differ- 
ences of  the  observed  readings  of  the  penstock  head  gage. 
The  fourth  column  gives  the  discharge  Q  through  the  wheel 


Interval 
of 
Time. 

Discharge 
Over 
Weir. 
Cub.  Feet 
per 
Minute. 

Fall  in 
Penstock. 
Feet. 

Flow 
through 
Wheel. 
Cub.  Feet 
per 
Minute. 

Theoretic 
Head  on     Horse  - 
Wheel,      power  of 
Feet.            the 
Water. 

Effective 
Horse- 
power of 
the 
Wheel. 

Efficiency 
of  the 
Wheel. 
Per  Cent. 

I7m 

to  i8m 

58- 

49 

+  0 

08 

59-77 

II 

41 

.290 

0-433 

33-6 

18 

to  19 

58. 

66 

+  0 

04 

59-30 

II 

36 

.274 

0.426 

33-4 

19 

tO   20 

58. 

49 

4-0 

03 

58.97 

II 

32 

.262 

0-433 

34-3 

22m 

to  23m 

58. 

05 

4-0 

13 

60.13 

10 

95 

•245 

0-437 

35-1 

23 

to  24 

58. 

05 

4-0 

07 

59-17 

IO 

86 

•  215 

0.441 

36.3 

24 

to  25 

57- 

88 

+  0 

05 

58.68 

10 

80 

.198 

0-437 

36.5 

found  as  above  explained.  The  fifth  column  is  the  mean 
head  h  on  the  wheel  during  the  minute,  as  derived  from 
the  observed  readings  of  head  and  tail  gage.  The  sixth 
column  is  found  by  formula  (137)2,  using  for  W  its  value 
•fawQ,  in  which  w  is  taken  at  62.4  pounds  per  cubic  foot. 
The  seventh  column  is  computed  from  formula  (140)2; 
and  the  last  column  is  found  by  dividing  the  numbers  in 
the  seventh  by  those  in  the  sixth  column. 

These  results  show  that  the  mean  efficiency  of  the  wheel 
and  shaft  under  the  conditions  stated  was  about  35  percent ; 
this  low  figure  being  due  to  the  circumstance  that  the  gate 
was  not  fully  opened.  It  is  also  seen  that  the  mean  effi- 
ciency of  the  second  set  is  2.2  percent  greater  than  that  of 
the  first  set;  this  is  due  to  the  lower  speed,  and  with  still 
lower  speeds  the  efficiency  was  found  to  be  lower,  so  that 
a  speed  of  about  535  revolutions  per  minute  gives  the 
maximum  efficiency. 

The  work  of  Francis  on  the  experiments  made  by  him  at 
Lowell,  Mass.,  will  always  be  a  classic  in  American  hy- 
draulic literature,  for  the  methods  therein  developed  for 
measuring  the  theoretic  power  of  a  waterfall,  and  the 


ART.  141  TESTS    OF   TURBINE    WHEELS  369 

effective  power  utilized  by  the  wheel,  are  models  of  careful 
and  precise  experimentation.*  In  determining  the  speed 
of  the  wheel  he  used  a  method  somewhat  different  from 
that  above  explained,  namely,  the  counter  attached  to  the 
shaft  was  connected  with  a  bell  which 'struck  at  the  com- 
pletion of  every  50  revolutions ;  the  observer  at  the  counter 
had  then  only  to  keep  his  eye  upon  the  watch,  and  to  note 
the  time  at  certain  designated  intervals — say  at  every 
sixth  stroke  of  the  bell.  The  number  of  revolutions  per 
second  was  then  obtained  by  dividing  the  number  of  revo- 
lutions in  the  interval  by  the  number  of  seconds,  as  deter- 
mined by  the  watch.  This  method  requires  a  stop-watch 
in  order  to  do  good  work,  unless  the  observer  be  very 
experienced,  as  an  error  of  one  second  in  an  interval  of  one 
minute  amounts  to  1.7  percent. 

At  Holyoke,  Mass.,  there  is  a  permanent  flume  for  test- 
ing turbines  arranged  with  a  weir  which  can  be  varied  up 
to  lengths  of  20  feet,  so  as  to  test  the  largest  wheels  which 
are  constructed.  As  the  expense  of  fitting  up  the  appa- 
ratus for  testing  a  large  turbine  at  the  place  where  it  is  to 
be  used  is  often  great,  it  is  sometimes  required  in  contracts 
that  the  wheel  shall  be  sent  to  a  place  where  a  special  outfit 
for  such  work  exists.  The  wheel  is  mounted  in  the  testing 
flume,  and  there,  by  the  methods  explained  in  the  preced- 
ing articles,  it  is  run  at  different  speeds  in  order  to  deter- 
mine the  speed  which  gives  the  maximum  efficiency  as  well 
as  the  effective  power  developed  at  each  speed.  As  the 
efficiency  of  a  turbine  varies  greatly  with  the  position  of 
the  gate  which  admits  the  water  to  it,  tests  are  made  with 
the  gate  fully  opened  and  at  various  partial  openings.  The 
results  thus  obtained  are  not  only  valuable  in  furnishing 
full  information  concerning  the  effective  power  and  effi- 
ciency of  the  wheel,  but  they  also  convert  the  turbine  into 
a  water  meter,  so  that  when  running  under  the  same  head 

*  Lowell  Hydraulic  Experiments,  ist  Edition,  1855;  4tn>  1883. 


Head 
in  Feet. 

Revolutions 
per 
Minute. 

Discharge. 
Cubic  Feet 
per  Second. 

Horse-power. 

Efficiency 
Per  Cent. 

17.  16 

63.5 

117.01 

172.57 

75.85 

17.27 

7O.O 

118.37 

177.41 

76.60 

17-33 

75-0 

"9-53 

178.63 

76.  II 

17-34 

80.0 

121.15 

178.32 

74.92 

17.21 

86.0 

122  .41 

178.57 

74-81 

17.21 

93-2 

124.74 

176.44 

72.54 

17.19 

100.  0 

127.73 

167.94 

67.51 

370  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

as  during  the  tests  the  quantity  of  water  which  passes 
through  it  can  at  any  time  be  closely  ascertained. 

The  following  gives  the  results  of  the  tests  of  an  80 -inch 
outward-flow  Boyden  turbine,  made  at  Holyoke  in  1885, 
the  gate  being  fully  opened  in  each  experiment.  The  heads 
in  the  second  column  were  derived  from  the  head  and  tail 

Number. 

21 
20 
19 

18 

i? 
16 

15 

race  gages,  these  being  arranged  so  that  one  observer  could 
directly  read  the  difference.  The  numbers  in  the  third 
column  were  found  by  dividing  the  total  number  of  revo- 
lutions during  the  experiment  by  its  length  in  minutes; 
those  in  the  fourth  by  the  weir  formula  (63) ;  those  in  the 
fifth  by  (140)  2  from  the  records  of  the  friction  dynamometer ; 
and  those  in  the  last  column  were  computed  by  (137)3.  It 
is  seen  that  the  discharge  always  increased  with  the  speed 
of  the  wheel,  and  the  reason  for  this  is  explained  in  Art.  166. 
The  maximum  efficiency  of  76.6  percent  occurred  at  70 
revolutions  per  minute;  and  for  100  revolutions  per  minute 
the  efficiency  was  lowered  to  67.5  percent,  notwithstanding 
that  the  quantity  of  water  passing  through  the  wheel  was 
much  greater. 

Prob.  141  a.  Compute  the  theoretic  horse-power  and  the  effi- 
ciency for  the  above  experiments,  Nos.  15  and  21,  on  the  large 
Boyden  outward-flow  turbine. 

Prob.  141&.  Compute  the  discharge  over  the  weir,  the  flow 
through  the  wheel,  the  theoretic  and  effective  horse-power, 
and  the  efficiency  for  the  tests  of  the  above  small  Eureka 
turbine  for  the  interval  from  3.24  to  3.25  P.M. 


ART.  142  FACTS    CONCERNING    WATER    POWER  371 


ART.  142.     FACTS  CONCERNING  WATER  POWER 

The  number  of  horse-powers  generated  by  water  wheels 
and  turbines  and  used  in  manufacturing  establishments  in 
the  United  States  was  i  130  431  in  1870,  i  225  379  in  1880, 
i  263  343  in  1890,  and  i  727  258  in  1900.  These  figures 
do  not  include  the  electric  power  derived  from  water,  which 
in  1900  was  probably  nearly  i  ooo  ooo  horse-powers.  Since 
1890  there  has  been  a  large  development  of  water  power  in 
connection  with  electric  light  and  trolley  service,  and  this 
development  promises  to  attain  great  proportions  during 
the  twentieth  century.  It  has  been  estimated  that  the 
rivers  of  the  United  States  can  furnish  about  200  ooo  ooo 
horse-powers,  so  that  the  possibilities  for  the  future  are 
almost  unlimited,  and  when  coal  becomes  very  high  in  price 
water  is  sure  to  take  the  place  of  steam. 

Water  power  is  sometimes  sold  by  what  is  called  the 
'  *  mill  power, ' '  which  may  be  roughly  supposed  to  be  such 
a  quantity  as  the  average  mill  requires,  but  which  in  any 
particular  case  must  be  defined  by  a  certain  quantity  of 
water  under  a  given  head.  *  Thus  at  Lowell  the  mill  power 
is  30  cubic  feet  per  second  under  a  head  of  25  feet,  which 
is  equivalent  to  85 . 2  theoretic  horse-power.  At  Minneapolis 
it  is  30  cubic  feet  per  second  under  22  feet  head,  or  75  theo- 
retic horse-power.  At  Holyoke  it  is  38  cubic  feet  per  second 
under  20  feet  head,  or  86.4  theoretic  horse-power.  This 
seems  an  excellent  way  to  measure  power  when  it  is  to  be 
sold  or  rented,  as  the  head  in  any  particular  instance  is  not 
subject  to  much  variation;  or  if  so  liable,  arrangements 
must  be  adopted  for  keeping  it  nearly  constant,  in  order 
that  the  machinery  in  the  mill  may  be  run  at  a  tolerably 
uniform  rate  of  speed.  Thus  nothing  remains  for  the  water 
company  to  measure  except  the  water  used  by  the  con- 
sumer. The  latter  furnishes  his  own  motor,  and  is  hence 
interested  in  securing  one  of  high  efficiency,  that  he  may 


372  WATER  SUPPLY  AND  WATER  POWER         CHAP,  xi 

derive  the  greatest  power  from  the  water  for  which  he  pays. 
The  perfection  of  American  turbines  is  undoubtedly  largely 
due  to  this  method  of  selling  power,  and  the  consequent 
desire  of  the  mill  owners  to  limit  their  expenditure  of  water. 
The  turbine  itself  when  tested  and  rated  becomes  a  meter 
by  which  the  company  can  at  any  time  determine  the  quan- 
tity of  water  that  passes  through  it. 

A  common  method  of  selling  the  power  which  is  gen- 
erated by  turbines  is  by  the  nominal  horse-power  of  the 
wheel  as  stated  in  the  catalog  of  the  manufacturer.  The 
seller  fixes  a  price  per  annum  for  one  horse-power  on  this 
basis,  and  the  buyer  furnishes  his,  own  wheel.  By  this 
method  no  controversy  can  arise  regarding  the  amount 
of  water  used,  for  the  purchaser  has  the  right  to  use  all 
that  can  pass  through  the  turbine.  This  method  cannot 
be  used  for  other  kinds  of  wheels,  because  the  gates  that 
regulate  the  flow  of  water  through  the  motor  is  not  a  part 
of  the  motor  itself. 

The  power  of  electric  generators  is  usually  expressed 
in  kilowatts.  One  English  horse-power  is  0.746  kilowatts, 
and  one  metric  horse-power  is  0.736  kilowatts.  One  kilo- 
watt is  1.340  English  horse-powers  or  1.360  metric  horse- 
powers. The  efficiency  of  a  good  electric  generator  is  about 
95  percent,  so  that  it  delivers  95  percent  of  the  work  im- 
parted to  it  by  the  turbine  wheel;  if  the  efficiency  of  this 
wheel  be  75  percent,  the  combined  efficiency  of  the  hydraulic 
and  electric  plant  is  71  percent.  Electric  power  is  often 
sold  by  the  kilowatt-hour,  this  being  measured  by  a  watt- 
meter. 

In  Art.  173  will  be  found  an  account  of  the  power  devel- 
opment at  Niagara  Falls  by  one  company,  its  plant  of 
105  ooo  electrical  horse-power  being  completed  at  the  close 
of  1903.  The  mean  discharge  of  the  Niagara  River  is  about 
280  ooo  cubic  feet  per  second,  and  the  fall  obtainable  by  the 
use  of  turbines  is  about  145  feet.  If  only  two-thirds  of  the 


ART.  142  FACTS    CONCERNING    WATER    POWER  373 

power  due  to  these  quantities  can  be  electrically  developed, 
the  number  of  horse-powers  obtainable  is  nearly  3^  millions. 
If  the  plans  of  other  companies  are  carried  out,  about 
500  ooo  horse-powers  will  probably  be  generated  at  these 
great  falls  by  the  year  1910.  If  this  process  of  power  de- 
velopment is  continued  during  the  following  years,  a  great 
diminution  in  the  quantity  of  water  passing  over  the  falls 
will  result. 

The  available  power  of  natural  waterfalls  is  very  great, 
but  it  is  probably  excee4ed  by  that  which  can  be  derived 
from  the  tides  and  waves  of  the  ocean.  Twice  every  day, 
under  the  attraction  of  the  sun  and  moon,  an  immense 
weight  of  water  is  lifted,  and  it  is  theoretically  possible  to 
derive  from  this  a  power  due  to  its  weight  and  lift.  Con- 
tinually along  every  ocean  beach  the  waves  dash  in  roar 
and  foam,  and  energy  is  wasted  in  heat  which  by  some 
device  might  be  utilized  in  power.  The  expense  of  deriv- 
ing power  from  these  sources  is  indeed  greater  than  that 
of  the  water  wheel  under  a  natural  fall,  but  the  time  may 
come  when  the  profit  will  exceed  the  expense,  and  then  it 
will  certainly  be  done.  Coal  and  wood  and  oil  may  become 
exhausted,  but  as  long  as  the  force  of  gravitation  exists, 
and  the  ocean  remains  upon  which  it  can  act,  power,  heat, 
and  light  can  be  generated  in  unlimited  quantities. 

Prob.  142a.  Deduce  the  simple  and  useful  rule  that  one 
cubic  foot  per  second  of  runoff  is  very  closely  equivalent  to  two 
acre-feet  per  day. 

Prob.  1426.  Find  the  theoretic  horse-power  of  a  plant  where 
1 200  cubic  feet  of  water  per  second  is  used  under  a  total  head 
of  49.5  feet.  If  the  efficiency  of  the  approaches  is  99  percent, 
the  efficiency  of  the  turbines  76  percent,  and  the  efficiency  of 
the  dynamos  96  percent,  what  power  in  kilowatts  is  delivered? 

Prob.  I42c.  What  is  the  theoretic  metric  horse-power  of  a 
plant  where  112  cubic  meters  of  water  per  second  are  used  under 
a  head  of  23.5  meters?  If  the  efficiencies  of  the  approaches. 


374  WATER  SUPPLY  AND  WATER  POWER        CHAP,  xi 

turbines,  and  electric  generators  are  98.5,  74.3,  and  97.5  percent 
respectively,  compute  the  number  of  metric  horse-powers  de- 
livered, and  also  the  power  in  kilowatts. 

Prob.  142<i.  When  a  turbine  is  tested  by  a  friction  dyna- 
mometer, show  that  its  power  in  kilowatts  is  o.ooio^NPl,  if  P 
be  the  load  on  the  brake  in  kilograms,  /  its  lever-arm  in  meters, 
and  N  the  number  of  revolutions  per  minute.  When  N  =  2oo, 
P  =  2$o  kilograms,  and  /  =  2.oi  meters,  what  electric  power  is 
delivered  by  a  dynamo  attached  to  the  turbine  when  the  effi- 
ciency of  the  dynamo  is  97.2  percent? 

Prob.  142e.  The  hectare-meter  is  a  convenient  unit  for  es- 
timating large  quantities  of  water  in  irrigation  and  water 
supply  work.  Show  that  one  hectare-meter  is  10  ooo  cubic 
meters.  Show  that  100  centimeters  of  rainfall  per  month  is, 
very  nearly,  0.004  cubic  meters  per  second  per  hectare. 


ART.  143  DEFINITIONS   AND    PRINCIPLES  375 


CHAPTER  XII 
DYNAMIC  PRESSURE  OF  WATER 

ART.  143.     DEFINITIONS  AND  PRINCIPLES 

The  pressures  exerted  by  moving  water  against  surfaces 
which  change  its  direction  or  check  its  velocity  are  called 
dynamic,  and  they  follow  very  different  laws  from  those 
which  govern  the  static  pressures  that  have  been  discussed 
and  used  in  the  preceding  chapters.  A  static  pressure  due 
to  a  certain  head  may  cause  a  jet  to  issue  from  an  orifice; 
but  this  jet  in  impinging  upon  a  surface  may  cause  a  dynamic 
pressure  less  than,  equal  to,  or  greater  than  that  due  to  the 
head.  A  static  pressure  at  a  given  point  in  a  mass  of  water 
is  exerted  with  equal  intensity  in  all  directions;  but  a 
dynamic  pressure  is  exerted  in  different  directions,  with 
different  intensities.  In  the  following  chapters  the  words 
static  and  dynamic  will  generally  be  prefixed  to  the  word 
pressure,  so  that  no  intellectual  confusion  may  result. 

The  dynamic  pressure  exerted  by  a  stream  flowing  with 
a  given  velocity  against  a  surface  at  rest  is  evidently  equal 
to  that  produced  when  the  surface  moves  in  still  water  with 
the  same  velocity.  This  principle  was  applied  in  Art.  40 
in  rating  the  current  meter,  the  vanes  of  which  move  under 
the  impulse  of  the  impinging  water.  The  dynamic  pressure 
exerted  upon  a  moving  body  by  a  flowing  stream  depends 
upon  the  velocity  of  the  body  relative  to  the  stream. 

The  ''impulse"  of  a  jet  or  stream  of. water  is  defined  as 
the  dynamic  pressure  which  it  is  capable  of  producing  in  the 
direction  of  its  motion  when  its  velocity  is  entirely  destroyed 


376 


DYNAMIC  PRESSURE  OF  WATER 


CHAP.  XII 


in  that  direction.  This  can  be  done  by  deflecting  the  jet 
normally  sidewise  by  a  fixed  surface;  if  the  surface  is 
smooth,  so  that  no  energy  is  lost  in  frictional  resistances, 
the  actual  velocity  remains  unaltered,  but  the  velocity  in 
the  original  direction  has  been  rendered  null.  In  Art.  29 
it  is  proved  that  the  theoretic  force  of  impulse  of  a  stream 
of  cross-section  a  and  velocity  v  is 


F  =  W-  =  wq- 
£  £ 

o  o 


(143) 


FIG.  143 


in  which  W  and  q  are  the  weight  and  volume  delivered  per 
second,  and  w  is  the  weight  of  one  cubic  unit  of  water.  This 
equation  shows  that  the  dynamic  pressure  that  may  be  pro- 
duced by  impulse  is  equal  to  the  static  pressure  due  to  twice 

the  head  corresponding  to  the  veloc- 
ity v.  It  would  then  be  expected 
that  if  two  equal  orifices  or  tubes  be 
placed  exactly  opposite,  as  in  Fig.. 
143,  and  a  loose  plate  be  placed  ver- 
tically against  one  of  them,  that  the 
dynamic  pressure  upon  the  plate 
caused  by  the  impulse  of  the  jet 
issuing  from  A  under  the  head  h 
would  balance  the  static  pressure  caused  by  the  head  2/z. 
This  conclusion  has  been  confirmed  by  experiment,  when 
the  tube  A  has  a  smooth  inner  surface  and  rounded  inner 
edges  so  that  its  coefficient  of  discharge  is  unity. 

The  reaction  of  a  jet  or  stream  is  the  backward  dynamic 
pressure,  in  the  line  of  its  motion,  which  is  exerted  against 
a  vessel  out  of  which  it  issues,  or  against  a  surface  away 
from  which  it  moves.  This  is  equal  and  opposite  to  the 
impulse,  and  the  equation  above  given  expresses  its  value 
and  the  laws  which  govern  it.  The  expression  for  the 
reaction  or  impulse  F  in  (143)  may  be  also  proved  as  fol- 
lows: The  definition  by  which  forces  are  compared  with 
each  other  is,  that  forces  are  proportional  to  the  accelera- 


ART.  143  DEFINITIONS   AND    PRINCIPLES  377 

tions  which  they  can  produce.  The  weight  W  if  allowed 
to  fall  acquires  the  acceleration  g\  the  force  F  which  can 
produce  the  acceleration  v  is  hence  related  to  W  and  g  by 
the  equation  F/W  =v/g,  and  accordingly  F  =W  .v/g. 

The  forces  of  impulse  and  reaction  do  not  really  exist 
in  a  stream  flowing  with  constant  velocity  and  direction, 
but  F  indicates  the  force  that  was  exerted  in  putting  the 
stream  into  motion  and  the  force  that  is  required  to  stop 
it.  If  the  direction  of  the  stream  be  changed  by  opposing 
obstacles,  the  impulse  and  reaction  produce  dynamic  pres- 
sure; if  in  making  this  change  the  absolute  velocity  is. 
retarded,  energy  is  converted  into  work.  Impulse  and. 
reaction  are  of  no  practical  value,  except  in  so  far  as  the 
resulting  dynamic  pressures  may  be  utilized  for  the  pro- 
duction of  work.  For  this  purpose  water  is  made  to  im- 
pinge upon  moving  vanes,  which  alter  both  its  direction 
and  velocity,  thus  producing  a  dynamic  pressure  P,  which, 
overcomes  in  each  second  an  equal  resisting  force  through 
the  space  u.  The  work  done  per  second  is  then  k  =Pu,  and 
it  is  the  object  in  designing  a  hydraulic  motor  to  make  this 
work  as  large  as  possible ;  for  this  purpose  the  most  advan- 
tageous values  of  P  and  u  are  to  be  selected. 

The  word  "impact"  is  sometimes  popularly  used  to- 
designate  impulse  or  pressure,  but  in  hydraulics  it  refers 
to  those  cases  where  energy  is  lost  in  eddies  and  foam,  as 
when  a  jet  impinges  into  water  or  upon  a  rough  plane  sur- 
face. Impact  is  not  denned  in  algebraic  terms,  but  the 
energy  lost  in  impact  may  be  so  denned  and  computed. 
When  the  energy  of  a  stream  of  water  is  to  be  utilized  T 
losses  due  to  impact  should  be  avoided.  Whenever  im- 
pact occurs  kinetic  energy  is  transformed  into  heat. 

Prob.  143.  If  a  jet  is  one  inch  in  diameter,  how  many  gallons 
per  second  must  it  deliver  in  order  that  its  impulse  may  be  100 
pounds  ? 


378 


DYNAMIC  PRESSURE  OF  WATER 


CHAP.  XII 


ART:  144.     EXPERIMENTS  ON  IMPULSE  AND  REACTION 

A  simple  device  by  which  the  dynamic  pressure  P  exerted 
upon  a  surface  by  the  impulse  and  reaction  of  a  jet  that 
glides  over  it  can  be  directly  weighed  is  shown  in  Fig.  144a. 

It  consists  merely  of  a  bent  lever 
supported  on  a  pivot  at  0,  and 
having  a  plate  A  attached  at 
the  lower  end  of  the  vertical  arm 
upon  which  the  stream  impinges, 
while  weights  applied  at  the  end 
of  the  other  arm  measure  the 
dynamic  pressure  produced  by 
the  impulse.  By  means  of  an  apparatus  of  this  nature, 
experiments  have  been  made  by  Bidone,  Weisbach,  and 
others,  the  results  of  which  will  now  be  stated. 

When  the  surface  upon  which  the  stream  impinges  is  a 
plane  normal  to  the  direction  of  the  stream,  as  shown  at  A, 
the  dynamic  pressure  P  closely  agrees  with  that  given  by 
the  theoretic  formula  for  F  in  the  last  article,  namely, 

v  ?;2 

(144) 


FIG.  144a 


being  about  2  percent  greater  according  to  Bidone,  and 
about  4  percent  less  according  to  Weisbach.  The  actual 
value  of  P  was  found  to  vary  somewhat  with  the  size  of 
the  plate,  and  with  its  distance  from  the  end  of  the  tube 
from  which  the  jet  issued. 

When  the  surface  upon  which  the  stream  impinges  is 
curved,  as  at  B,  or  so  arranged  that  the  water  is  turned  back- 
ward from  the  surface,  the  value  of  the  dynamic  pressure 
P  was  found  to  be  always  greater  than  the  theoretic  value, 
and  that  it  increased  with  the  amount  of  backward  incli- 
nation. When  a  complete  reversal  of  the  original  direction 
of  the  water  was  obtained,  as  at  C,  it  was  found  that  P,  as 


ART.  144        EXPERIMENTS  ON  IMPULSE  AND  REACTION  379 

measured  by  the  weights,  was  nearly  double  the  value  of 
that  against  the  plane.  This  is  explained  by  stating  that 
as  long  as  the  direction  of  the  flow  is  toward  the  surface 
the  dynamic  pressure  of  its  impulse  is  exerted  upon  it; 
when  the  water  flows  backward  away  from  the  surface 
the  dynamic  pressure  due  to  both  impulse  and  reaction 
is  exerted  upon  it.  The  sum  of  these  is 

P=F  +  F  =  2W-=4wa— 
g  2g 

which  agrees  with  the  results  experimentally  obtained. 

An  experiment  by  Morosi*  shows  clearly  .that  the  dy- 
namic pressure  against  a  surface  may  be  increased  still 
further  by  the  device  shown  in  Fig.  1446,  where  the  stream 
is  made  to  perform  two  complete  reversals  upon  the  sur- 
face. He  found  that  in  this  case  the  value 
of  the  dynamic  pressure  was  3.32  times  as 
great  as  that  against  a  plane,  for  P  =  3«32F, 
whereas  theoretically  the  3.32  should  be  4.  In 
this  case,  as  in  those  preceding,  the  water  in 
passing  over  the  surface  loses  energy  in  fric- 
tion and  foam,  so  that  its  velocity  is  dimin- 
ished, and  it  should  hence  be  expected  that  the  experimental 
values  of  the  dynamic  pressures  would  be  less  than  the  theo- 
retic values,  as  in  general  they  are  found  to  be. 

While  the  experiments  here  briefly  described  thoroughly 
confirm  the  results  of  theory,  they  further  show  it  is  the 
change  in  direction  of  the  velocity  when  in  contact  with  the 
surface  which  produces  the  dynamic  pressure.  If  the  stream 
strikes  normally  against  a  plane,  the  direction  of  its  ve- 
locity is  changed  90°,  and  this  is  the  same  as  the  entire 
destruction  of  the  velocity  in  its  original  direction,  so  that 
the  dynamic  pressure  P  should  agree  with  the  impulse  F. 
This  important  principle  of  change  in  direction  will  be 
theoretically  exemplified  later. 

*  Ruhlman's  Hydromechanik  (Hannover,  1879),  p.  586. 


380 


DYNAMIC  PRESSURE  OF  WATER 


CHAP.  XII 


The  dynamic  pressure  which  is  produced  by  the  direct 
reaction  of  a  stream  of  water  when  issuing  from  a  vertical 
orifice  in  the  side  of  a  vessel  was  measured 
by  Ewart  with   the  apparatus  shown  in 
Fig.  144<;,   which  will   be    readily  under- 
stood   without     a    detailed     description. 
The  discussion  of  these  experiments  made 
by  Weisbach  *  shows  that  the  measured 
values  of  P  were  from  2  to  4  percent  less 
FIG.  144c         than  the  theoretic  value  F  as  given  by 
(144),  so  that  in  this  case  also  theory  and  observation  are 
in  accordance. 

An  experiment  by  Unwin,f  illustrated  in  Fig.  144d,  is 
very  interesting,  as  it  perhaps  explains  more  clearly  than 
formula  (143)  why  it  is  that  the  dynamic  pressure  due  to 
impulse  is  double  the  static  pres- 
sure.  Two  vessels  having  converg- 
ing tubes  of  equal  size  were  placed 
so  that  the  jet  from  A  was  directed 
exactly  into  B.  The  head  in  A 
was  kept  uniform  at  20^  inches, 
when  it  was  found  that  the  water  FlG-  U4d 

in  B  continued  to  rise  until  a  head  of  18  inches  was  reached. 
All  the  water  admitted  into  A  was  thus  lifted  in  B  by  the 
impulse  of  the  jet,  with  a  loss  of  2\  inches  of  head,  which 
was  caused  by  foam  and  friction.  If  such  losses  could  be 
entirely  avoided,  the  water  in  B  might  be  raised  to  the  same 
level  as  that  in  A.  In  the  case  shown  in  the  figure  where 
the  water  overflows  from  B,  the  impulse  of  the  jet  has  not 
only  to  overcome  the  static  pressure  due  to  the  head  h,  but 
also  to  furnish  the  dynamic  pressure  equivalent  to  a  second 
head  h  in  order  to  raise  the  water  through  that  height. 
But  the  level  in  B  can  never  rise  higher  than  in  A,  for  the 

*  Theoretical  Mechanics,  Coxe's  translation,  vol.  i,  p.  1004. 
f  Encyclopaedia  Britannica,  9th  Edition,  vol.  12,  p.  467. 


ART-  145  SURFACES  AT  REST  381 

velocity-head  of  the  jet  cannot  be  greater  than  that  of  the 
static  head  which  generates  it. 

Prob.  144a.  In  Fig.  144a  the  diameter  of  the  tube  is  i  inch, 
and  it  delivers  0.3  cubic  feet  per  second.  Compute  the  theoretic 
dynamic  pressure  against  the  plane. 

Prob.  1446.  Accepting  as  an  experimental  fact  that  the  force 
of  impulse  or  reaction  is  double  the  static  pressure,  show  that 
the  theoretic  velocity  of  flow  is  \/2gh. 

ART.  145.     SURFACES  AT  REST 

Let  a  jet  of  water  whose  cross-section  is  a  impinge  in 
permanent  flow  with  the  uniform  velocity  v  upon  a  surface 
at  rest.  Let  the  surface  be  smooth,  so  that  no  resisting 
forces  of  friction  exist,  and  let  the  stream  be  prevented  from 
spreading  laterally.  The  water  then  passes  over  the  surface, 
and  leaves  it  with  the  original  velocity  v,  producing  upon 


D 


it  a  dynamic  pressure  whose  value  depends  upon  its  change 
of  direction.  At  B  in  Fig.  145a  the  stream  is  deflected 
normal  to  its  original  direction,  and  at  D  a  complete  rever- 
sal is  effected.  Let  6  be  the  angle  between  the  initial  and 
final  directions,  as  shown.  It  is  required  to  determine  the 
dynamic  pressure  exerted  upon  the  surface  in  the  same 
direction  as  that  of  the  jet.  In  the  above  figures,  as  in 
those  that  follow,  the  stream  is  supposed  to  lie  in  a  horizon- 
tal plane,  so  that  no  acceleration  or  retardation  of  its 
velocity  will  be  produced  by  gravity. 

The  stream  entering  upon  the  surface  exerts  its  impulse 
F  in  the  same  direction  as  that  of  its  motion;  leaving 
the  surface  it  exerts  its  reaction  F  in  opposite  direc- 


382  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

tion  to  that  of  its  motion.     Let  P  be  the  dynamic  pressure 

thus  produced  in  the  di- 
*fi*        F\\*»  \    rection  of  the  initial  mo- 

the  reaction  F  in  the  same 
direction.     Then 
F  : 

FIG.  1456 
and  inserting  for  F  its  value  as  given  by  (143), 

P  =  (i  -cos6)W-  (145)t 

which  is  the  formula  for  the  dynamic  pressure  in  the  direc- 
tion of  the  impinging  jet.  If  in  this  6=0°,  the  stream 
glides  along  the  surface  without  changing  its  direction,  and 
P  becomes  zero;  if  6  is  90°,  the  dynamic  pressure  is 

v 

P  =  F  =  W— 
g 

and  if  6  becomes  180°  a  complete  reversal  of  direction  is 
obtained,  and  the  dynamic  pressure  that  is  exerted  by  the 
jet  against  the  surface  is 

v 

g 

These  theoretic  conclusions  agree  with  the  experimental 
results  described  in  the  last  article.  In  the  deduction  of 
(145)!  the  angle  6  has  been  regarded  as  less  than  90°,  but 
the  same  formula  results  if  6  be  considered  greater  than 
90°,  since  then  the  sign  of  Fx  is  positive. 

The  resultant  dynamic  pressure  exerted  upon  the  sur- 
face is  found  by  combining  by  the  parallelogram  of  forces 
the  impulse  F  and  the  equal  reaction  F.  In  Fig.  1456  it 
is  seen  that  this  resultant  bisects  the  angle  180  —  6,  and 
that  its  value  is 

P'  =  2F  cos  ^(180  -  0)  =  2  sin  £0 .  W- 

o 


ART.  145  SUKFACES   AT   REST  383 

It  is  usually,  however,  more  important  to  ascertain  the  pres- 
sure in  a  given  direction  than  the  result- 
ant. This  can  be  found  by  taking  the 
component  of  the  resultant  in  that  direc- 
tion, or  by  taking  the  algebraic  sum  of  the 
components  of  the  initial  impulse  and  the 
final  reaction. 

To  find  the  dynamic  pressure  P  in  a 
direction  which  makes  an  angle  a  with 
the  entering  and  the  angle  0  with  the  departing  stream,  the 
components  in  that  direction  are 

Pl=F  cosa         P2  =  —F  cos# 
and  the  algebraic  sum  of  these  two  components  is 

P  =F(cosa  -cosfl)  =  (cosa  -costf)  W-         (145)2 

o 

This  becomes  equal  to  F  when  a  =o  and  6  =90°,  as  at  B  in 
Fig.  145a,  and  also  when  a  =  90°  and  6  =  180°.  When  a:  =0° 
and  6  =  180°  the  entering  and  departing  streams  are  parallel, 
as  at  D  in  Fig.  145a,  so  that  the  value  of  P  is  2F,  which  in 
this  case  is  the  same  as  the  resultant  pressure. 

The  formulas  here  deduced  are  entirely  independent  of 
the  form  of  the  surface,  and  of  the  angle  with  which  the  jet 
enters  upon  it.  It  is  clear,  however,  if,  as  in  the  planes  in 
Fig.  145a,  this  angle  is  such  as  to  allow  shock  to  occur,  that 
foam  and  changes  in  cross-section  may  result  which  will 
cause  energy  to  be  dissipated  in  heat.  These  losses  will 
diminish  the  velocity  v  and  decrease  the  theoretic  dynamic 
pressure.  These  effects  cannot  be  formulated,  but  it  is  a 
general  principle,  which  is  confirmed  by  experiment,  that 
they  may  be  largely  avoided  by  allowing  the  jet  to  impinge 
tangentially  upon  the  surface. 

In  all  the  foregoing  formulas  the  weight  W  which  im- 
pinges upon  the  surface  per  second  is  the  same  as  that  which 


384 


DYNAMIC  PRESSURE  OF  WATER 


CHAP.  XII 


issues  from  the  orifice  or  nozzle  that  supplies  the  stream,  or 

W  =  wq  =  wav 

To  find  W  it  is  hence  necessary  to  use  the  methods  of  the 
preceding  chapters  to  determine  either  the  discharge  q  or 
the  mean  velocity  v. 

Prob.  145.  If  F  is  10  pounds,  a  =  o°,  and  6  =  60°,  show  that 
the  pressure  parallel  to  the  direction  of  the  jet  is  5  pounds,  that 
the  pressure  normal  to  that  direction  is  8.66  pounds,  and  that 
the  resultant  dynamic  pressure  is  10  pounds. 


ART.  146.     IMMERSED  BODIES 

When  a  body  is  immersed  in  a  flowing  stream,  or  when 
it  is  moved  in  still  water,  so  that  filaments  are  caused  to 
change  their  direction,  a  dynamic  pressure  is  exerted  by 

^^_ „  _,^ -^^^      the     stream     or 

wOfjJT    ^^     ^W)^^   ^^S  overcome  by  the 

body.     It   is   to 
be  inferred  from 
what    has    pre- 
ceded that  the  dynamic  pressure  in  the  direction  of  the 
motion  is  proportional  to  the  force  of  impulse  of  a  stream 
which  has  a  cross-section  equal  to  that  of  the  body,  or 

P  =  m .  wa — 

in  which  m  is  a  number  depending  upon  the  length  and 
shape  of  the  immersed  portion,  and  whose  value  is  2  for  a 
jet  impinging  normally  upon  a  plane. 

Experiments  made  upon  small  plates  held  normally  to 
the  direction  of  the  flow  show  that  the  value  of  m  lies  be- 
tween 1.25  and  1.75,  the  best  determinations  being  near  1.4 
and  1.5.  It  is  to  be  expected  that  the  dynamic  pressure  on 
a  plate  in  a  stream  would  be  less  than  that  due  to  the  im- 
pulse of  a  jet  of  the  same  cross-section,  as  the  filaments  of 


ART.  146  IMMERSED  BODIES  385 

water  near  the  outer  edges  are  crowded  sideways,  and  hence 
do  not  impinge  with  full  normal  effect,  and  the  above  results 
confirm  this  supposition.  The  few  experiments  on  record 
were  made  with  small  plates,  mostly  less  than  2  square  feet 
area,  and  they  seem  to  indicate  that  m  is  greater  for  large 
surfaces  than  for  small  ones. 

The  determination  of  the  dynamic  pressure  upon  the 
end  of  an  immersed  cylinder  or  prism  is  difficult  because 
of  the  resisting  friction  of  the  sides;  but  it  is  well  ascer- 
tained to  be  less  than  that  upon  a  plane  of  the  same  area, 
and  within  certain  limits  to  decrease  with  the  length.  For 
a  conical  or  wedge-shaped  body  the  dynamic  pressure  is 
less  than  that  upon  the  cylinder,  and  it  is  found  that  its 
intensity  is  much  modified  by  the  shape  of  the  rear  surface. 

When  a  body  is  so  shaped  as  to  gradually  deflect  the 
filaments  of  water  in  front,  and  to  allow  them  to  gradually 
close  in  again  upon  the  rear,  the  impulse  of  the  front  fila- 
ments upon  the  body  is  balanced  by  the  reaction  of  those  in 
the  rear,  so  that  the  resultant  dynamic  pressure  is  zero. 
The  forms  of  boats  and  ships  should  be  made  so  as  to  secure 
this  result,  and  then  the  propelling  force  has  only  to  over- 
come the  fractional  resistance  of  the  surface  upon  the  water. 

The  dynamic  pressure  produced  by  the  impulse  of  ocean 
waves  striking  upon  piers  or  lighthouses  is  often  very  great. 
The  experiments  of  Stevenson  on  Skerry vore  Island,  where 
the  waves  probably  acted  with  greater  force  than  usual, 
showed  that  during  the  summer  months  the  mean  dynamic 
pressure  per  square  foot  was  about  600  pounds,  and  during 
the  winter  months  about  2100  pounds,  the  maximum  ob- 
served value  being  6100  pounds.  At  the  Bell  Rock  light- 
house the  greatest  value  observed  was  about  3000  pounds 
per  square  foot.  The  observations  were  made  by  allow- 
ing the  waves  to  impinge  upon  a  circular  plate  about  6 
inches  in  diameter,  and  the  pressure  produced  was  regis- 


386  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xir 

tered  by  the  compression  of  a  spring.  Such  high  unit-pres- 
sures do  not  probably  act  upon  large  areas  of  masonry 
exposed  to  wave  action.* 

Prob.  146.  Compute  the  probable  dynamic  pressure  upon  a 
surface  i  foot  square  when  immersed  in  a  current  whose  velocity 
is  8  feet  per  second,  the  direction  of  the  current  being  normal 
to  the  surface. 

ART.  147.     CURVED  PIPES  AND  CHANNELS 

The  dynamic  pressures  discussed  in  the  preceding  article 
have  been  those  caused  by  jets,  or  isolated  streams,  of 
water.  There  is  now  to  be  considered  the  case  of  dynamic 
pressures  caused  by  streams  flowing  in  pipes,  conduits,  or 
channels  of  any  kind;  these  are  sometimes  called  limited 
or  bounded  streams,  the  boundary  being  the  surface  whose 
cross-section  is  the  wetted  perimeter.  When  such  a  stream 
is  straight  and  of  uniform  section,  and  all  its  filaments  move 
with  the  same  velocity  v,  the  impulse,  or  the  pressure  which 
it  can  produce,  is  the  quantity  F  given  by  the  general  ex- 


FIG.  147a 

pression  in  Art.  143;  under  these  conditions  it  exerts  no 
dynamic  pressure,  but  if  a  body  be  immersed  and  held 
stationary,  pressure  is  produced  upon  it.  If  its  direction 
changes  in  an  elbow  or  bend,  pressure  upon  the  bounding 
surface  is  produced;  if  its  cross-section  increases  or  de- 
creases, pressure  is  developed  or  absorbed. 

*  Cooper,  on  Ocean  Waves,  in  Transactions  American  Society  of  Civil 
Engineers,  1896,  vol.  36,  p.  150. 


ART.  147  CURVED    PlPES   AND    CHANNELS  387 

The  resultant  dynamic  pressure  Pf  upon  a  curved  pipe 
which  runs  full  of  water  with  the  uniform  velocity  v  de- 
pends upon  the  angle  6  between  the  initial  and  final  direc- 
tions, and  must  be  the  same  as  that  produced  upon  a  sur- 
face by  an  impinging  jet  which  passes  over  it  without 
change  in  velocity.  The  formula  of  Art.  145  then  directly 
applies,  and 

P'  =2  sini0.F  =  2  sinitf.T^- 

o 

If  0  =0°,  there  is  no  bend,  and  Pf  =o  ;  if  6  =  180°,  the  direc- 
tion of  flow  is  reversed,  and  P'  =  2p.  If  the  direction  is 
twice  reversed,  as  at  C  in  Fig.  147  a,  the  value  of  6  is  360°,  and 
the  resultant  is  the  initial  impulse  F  minus  the  final  reac- 
tion F,  or  simply  zero  ;  in  this  case,  however,  there  may  be 
a  couple  which  tends  to  twist  the  pipe,  unless  the  impulse 
and  reaction  lie  in  the  same  line. 

The  dynamic  pressure  developed  in  a  unit  of  length  of 
the  curve  will  now  be  found.  Let  the  pipe  at  A  in  Fig.  147  a 
have  the  length  dl,  and  let  0  be  nearly  o°,  so  that  its  value  is 
the  elementary  angle  dd.  Then  in  the  above  formula  Pr 
becomes  the  elementary  radial  pressure  $Plt  and 


Now  since  dd  =  dl/R,  where  R  is  the  radius  of  the  curve, 
the  dynamic  pressure  developed  in  the  distance  dl  is  Fdl/R, 
and  that  for  a  unit  of  length  is  F/R.  Accordingly,  by 
Art.  144,  this  pressure  is 

F 


The  unit-pressure  pf  is  found  by  dividing  Pl  by  a,  and  the 
corresponding  head  ht  is  found  by  dividing  pf  by  w\  hence 

2WV2  2  V* 

p'  = —  and  »i  — 5  — 

R    2g  R2g 

are  the  values  for  one  unit  of  length  of  the  curve. 


388 


DYNAMIC  PRESSURE  OF  WATER 


CHAP.  XII 


The  dynamic  pressure-head  h1  is  developed  in  every  unit 
of  length  of  the  pipe.  It  is  not  known  how  these  influence 
the  static  pressure  or  how  they  affect  piezometers.  Nor 
is  it  known  whether  they  combine  so  that  the  dynamic 
pressure  becomes  greater  with  the  distance  from  the  be- 
ginning of  the  curve.  Undoubtedly,  however,  a  part  of 
HI  is  expended  in  causing  cross-currents  whereby  impact 
results  and  some  of  the  static  head  is  lost.  This  loss  should 
be  proportional  to  h1  and  proportional  to  the  length  /  of 
the  curve,  or,  if  d  is  the  diameter  of  the  pipe 


in  which  the  curvature  factor  /A  depends  upon  the  ratio 
R/d.  This  investigation  appears  to  indicate  that  pipes 
of  the  same  diameter  and  of  different  curvatures  give  the 
same  loss  of  head,  if  the  central  angle  is  the  same;  but,  as 
seen  in  Art.  87,  certain  experiments  seem  to  point  to  the 
conclusion  that  the  loss  per  linear  unit  is  greatest  in  the 
pipe  having  the  longest  radius. 

The  same  reasoning  applies  approximately  to  the 
curves  of  conduits,  canals,  and  rivers.  In  any  length  I 
there  exists  a  radial  dynamic  pressure  Plt  acting  toward 

the  outer  bank  and  causing; 
currents  in  that  direction, 
which,  in  connection  with 
the  greater  velocity  that 
naturally  there  exists,  tends 
to  deepen  the  channel  on 
that  side.  This  may  help 
to  explain  the  reason  why 
FlG  147&  rivers  run  in  winding  courses. 

At   first   the   curve  may  be 

slight,  but  the  radial  flow  due  to  the  dynamic  pressure  causes 
the  outer  bank  to  scour  away;  this  increases  the  velocity 


ART.  147         CURVED  PlPES  AND  CHANNELS  389 

v2  and  decreases  v^  (Fig.  1476),  and  this  in  turn  hastens 
the  scour  on  the  outer  and  allows  material  to  be  deposited 
on  the  inner  side.  Thus  the  process  continues  until  a 
state  of  permanency  is  reached,  and  then  the  existing 
forces  tend  to  maintain  the  curve.  The  cross-currents 
which  the  radial  pressure  produces  have  been  actually 
seen  in  experiments  devised  by  Thomson,*  and  it  is  found 
that  they  move  in  the  manner  shown  in  the  above  figure, 
the  motion  toward  the  outer  bank  being  in  the  upper  part 
of  the  section,  while  along  the  wetted  perimeter  they  flow 
toward  the  inner  bank.  When  the  slope  is  small  and  the 
mean  velocity  low  the  influence  of  the  cross-currents  is. 
relatively  greater  than  for  higher  slopes,  and  this  is  prob- 
ably one  of  the  reasons  why  the  sharpest  curves  are  found 
in  streams  of  slight  slope.  Perhaps  another  reason  for  this 
is  that  at  very  low  velocities  the  law  of  flow  is  different, 
the  head  varying  as  the  first  power  of  the  velocity 
(Art.  116),  and  the  energy  being  expended  in  friction  along 
the  banks  instead  of  in  impact. 

The  elevation  of  the  water  on  the  outer  bank  of  a  bend 
is  higher  than  on  the  inner.  This  is  only  a  partial  conse- 
quence of  the  radial  dynamic  pressure,  as  in  straight 
streams  it  is  also  seen  that  the  water  surface  is  curved, 
the  highest  elevation  being  where  the  velocity  is  greatest. 
In  this  case  cross-currents  are  also  observed  which  move 
near  the  surface  toward  the  center  of  the  stream,  and  near 
the  bottom  toward  the  banks,  their  motion  being  due  to 
the  disturbance  of  the  static  pressure  consequent  upon 
the  varying  water  level. 

Prob.  147.  The  mean  velocity  in  a  pipe  is  9  feet  per  second. 
If  it  be  laid  on  a  curve  of  3  feet  radius,  show  that  the  dynamic 
pressure-head  for  each  foot  in  length  of  the  pipe  is  0.84  feet. 
If  the  radius  of  the  curve  be  6  feet,  what  is  the  dynamic  pressure- 
head?  What  is  the  dynamic  pressure-head  for  each  case  when 
the  mean  velocity  is  3  feet  per  second? 

*  Proceedings  Royal  Society  of  London,  1878,  p.  356. 


390  DYNAMIC  PRESSURE  OF  WATER 


ART.  148.     WATER  HAMMER  IN  PIPES 


CHAP.  XII 


When  a  valve  in  a  pipe  is  closed  while  the  water  is 
flowing  the  velocity  of  the  water  is  retarded  as  the  valve 
descends  and  thus  a  dynamic  pressure  is  produced.  When 
the  valve  is  closed  quickly  this  dynamic  pressure  may  be 
much  greater  than  that  due  to  the  static  pressure,  and  it 
is  then  called  " water  hammer"  or  " water  ram."  Pipes 
have  often  been  known  to  burst  under  this  cause,  and  hence 
the  determination  of  the  maximum  dynamic  pressure  of 
the  water  hammer  is  a  matter  of  importance.  Fig.  148a 
illustrates  the  phenomena  of  water  hammer  for  the  closing 
of  a  valve  at  the  end  of  a  pipe  where  the  water  issues 
through  a  nozzle.  At  the  entrance  there  is  supposed  to 


be  a  gage  which  registers  the  static  unit-pressure  plt  while 
the  flow  is  in  progress,  and  the  static  unit-pressure  ^>0when 
there  is  no  flow.  The  abscissas  represent  time,  and  at  B 
the  valve  begins  to  close.  After  a  short  interval  of  time 
BC  the  gage  registers  the  unit-pressure  Cc\  after  another 
short  interval  the  unit-pressure  has  decreased  to  Dd,  and 
a  series  of  oscillations  follows  until  finally  the  disturbance 
ceases.  A  diagram  of  this  kind  may  be  autographically 
drawn  by  suitable  mechanism  connected  with  the  pressure 
gage,  and  such  were  made  in  the  experiments  conducted 
by  Carpenter,*  as  also  in  those  of  Fletcher,  f 

*  Transactions  American  Society  of  Mechancial  Engineers,  1894,  vol.  15, 
p.  150.        f  Engineering  News,  1898,  vol.  39,  p.  323. 


ART.  148  WATER  HAMMER  IN  PIPES  391 

Let  P  represent  the  excess  of  maximum  dynamic  unit- 
pressure  over  the  static  unit-pressure  when  there  is  no  flow; 
that  is,  the  difference  of  the  ordinates  Cc  and  Ee.  This 
is  the  excess  unit-pressure  due  to  the  water  hammer  and 
it  is  required  to  determine  an  expression  for  its  value. 
It  is  first  to  be  noted  that  the  actual  dynamic  unit-pressure 
produced  by  the  retardation  of  the  velocity  is  the  difference 
of  the  ordinates  Cc  and  Bb  and  this  difference  is  p  +  pQ  —  p^ 
The  dynamic  pressure  on  the  area  a  of  the  cross-section 
of  the  pipe  is  then  (p  +  po—pja,  and  for  brevity  this  may 
be  represented  by  P.  If  this  pressure  be  regarded  as 
varying  uniformly  from  o  up  to  P  during  the  time  t  in 
which  the  valve  closes,  its  mean  value  is  \P  and  its  total 
impulse  during  this  time  is  \Pt.  If  /  be  the  length  of  the 
pipe,  w  the  weight  of  a  cubic  unit  of  water,  and  v  the  veloc- 
ity during  the  flow,  the  total  weight  of  water  in  the  pipe 
is  wal  and  its  impulse  is  wal.v/g.  Equating  these  ex- 
pressions of  the  impulse  there  is  found  P  =  2wal/gt,  and 
replacing  P  by  its  value,  there  results 

2Wl 

P=-gV  +  Pi-p*  (148), 

as  the  excess  dynamic  unit-pressure  due  to  closing  the  valve 
in  the  time  t.  This  formula,  having  been  deduced  with- 
out considering  the  fact  that  time  is  required  for  the  trans- 
mission of  stress  through  water,  cannot  be  regarded  as 
applicable  to  all  cases. 

In  Art.  6  it  was  shown  that  the  velocity  with  which 
any  disturbance  is  propagated  through  water  is  about 
4670  feet  per  second,  and  this  velocity  may  be  represented 
by  u.  Now  let  the  pipe  of  length  I  have  an  open  valve  at 
the  end,  and  let  the  water  be  flowing  through  every  sec- 
tion with  the  velocity  v.  Then  the  time  l/u  must  elapse 
after  the  valve  begins  to  close  before  the  velocity  begins 
to  be  checked  at  the  upper  end  of  the  pipe,  and  the  further 
time  of  l/u  must  elapse  before  the  pressure  due  to  this 


392  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

retardation  can  be  transmitted  back  to  the  valve.  The 
total  time  2l/u  is  then  required  before  the  gage  at  the 
valve  can  indicate  the  pressure  due  to  the  retardations 
of  the  velocity  in  the  length  /.  Hence,  if  the  time  in  which 
the  valve  closes  be  equal  to  or  less  than  2l/u,  the  time  t 
in  the  above  formula  is  to  be  replaced  by  2l/u,  and  thus 

wu 

P=—v  +  Pi-p«  (148)a 

& 

is  the  maximum  excess  dynamic  unit-pressure  that  can 
occur  in  the  pipe.  This  depends  upon  the  velocity  of  the 
water  and  upon  the  initial  and  final  static  pressures. 

The  subject  of  water  hammer  in  pipes  is  one  of  the 
most  difHcult  in  hydromechanics,  and  the  above  investi- 
gation cannot  be  regarded  as  final.  Formula  (148)i  is 
probably  correct  only  for  a  certain  law  of  valve  closing. 
Formula  (148)2,  however,  is  certainly  correct,  for  it  may 
be  proved  by  other  methods,  one  of  which  is  as  follows: 
When  the  water  is  in  motion  the  kinetic  energy  in  a  length 
dl  Sit  the  gage  is  wa§l.v2/2g;  when  it  is  brought  to  rest 
under  the  unit-stress  s  its  stress  energy  is  adl.sz/2E,  if  E 
be  the  modulus  of  elasticity  of  the  water.*  Equating 
these  expressions,  and  substituting  p  +  pi—po  for  s,  there 
results  for  the  excess  dynamic  unit-pressure 


and  this  reduces  to  (148)2  if  E  be  replaced  by  wu*/g,  which 
is  its  value  according  to  formula  (6). 

When  v  is  in  feet  per  second,  and  pQ,  plf  and  p  are  in 
pounds  per  square  inch,  these  formulas  become 

p=o.o2>j(l/t)v  +  p1  -p0         p=62,v  +  p1  -p0       (148), 

the  first  of  which    is    to    be  used  when  t  is  greater  than 
o.  00042  8/  and  the  second  when  t  is  equal  to  or  less  than  it, 

*  Merriman's  Mechanics  of  Materials  (New  York,  ^1901),  p.  202. 


ART.  148  WATER   HAMMER   IN    PlPES  393 

/  being  in  feet.     From  the  first  of  these  formulas  the  value 
of  /,  when  p  =  o,  is  found  to  be 

Iv 


which  is  the  time  of  valve  closing  in  order  that  there  may 
be  no  water  hammer.  For  example,  let  p0  be  83  and  p1 
be  58  pounds  per  square  inch,  /  be  1903  feet,  and  v  be  5  feet 
per  second,  then  t  is  10.3  seconds.  To  prevent  the  effects 
of  water  hammer,  it  is  customary  to  arrange  valves  so  that 
they  cannot  be  closed  very  quickly,  and  the  last  formula 
furnishes  the  means  of  estimating  the  time  required  in 
order  that  no  excess  of  dynamic  pressure  over  the  static 
pressure  p0  may  occur. 

The  elaborate  experiments  of  Joukowsky  at  Moscow  in 
1898*  have  fully  confirmed  the  truth  of  formula  (148)2. 
Horizontal  pipes  of  2,  4,  and  6  inches  diameter,  with  lengths 
of  2494,  1050,  and  1066  feet,  were  used,  and  the  valve  at 
the  end  was  closed  in  0.03  seconds.  Ten  autographic  re- 
cording gages  were  placed  along  the  length  of  a  pipe,  and 
it  was  found  that  substantially  the  same  dynamic  pressure 
was  produced  at  each,  but  that  the  time  length  of  a  wave 
was  the  shorter  the  further  the  distance  of  a  gage  from  the 
valve;  this  wave  length  is  shown  in  the  above  figure  by 
the  distance  BD.  The  following  is  a  comparison  of  the 

For  the  4-inch  pipe:  For  the  6-inch  pipe: 

Velocity.  Observed.  Computed.  Velocity.  Observed.  Computed. 

0.5  3i  3i  0.6  43  38 

1.9  115  118  1.9  106  118 

2.9  168  183  3.0  173  189 

4-i  232  258  5.6  369  353 

9.2  519  580  7.5  426  472 

observed  values  of  p  +  p0  —p^  for  a  few  of  these  experiments 
with  the  values  computed  from  63^.  It  is  seen  that  the 
observed  are  less  than  the  computed  values  except  in  one 

*  Stoss  in  Wasserleitungsrohren,  St.  Petersburg,  1900.     Translation  from 
the  Memoirs  of  the  St.  Petersburg  Academy  of  Sciences. 


394  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

instance,  and  Joukowsky  concludes  that,  owing  to  the  in- 
fluence of  the  metal  of  the  pipes,  the  velocity  u  with  which 
stress  is  transmitted  in  the  water  is  about  4200  instead  of 
4670  feet  per  second.  This  conclusion  may  be  applied  in 
practice  by  using  597;  instead  of  6$v  in  (148)3. 

If  the  pipe  be  a  compound  one  (Art.  96)  the  above  for- 
mulas also  hold  for  any  cross-section,  if  v  be  the  velocity 
and  p0  and  pt  the  static  and  hydraulic  pressures  at  that 
section,  and  /  be  the  length  of  the  pipe  from  the  valve  to 
the  reservoir.  In  a  system  of  pipes  having  diversions  and 
branches  (Art.  100),  it  is  often  difficult  to  tell  what  length 
to  use  for  /  in  (148)j.  For  a  house-service  pipe  connected 
with  a  street  main,  /  is  usually  the  length  from  the  valve  to 


X,     :        "        • 


FIG.  1486 


the  main.  In  computing  the  thickness  of  water  pipes,  it 
is  customary  to  allow  100  pounds  per  square  inch  for  the 
influence  of  water  hammer.  This  is  equivalent,  if  p{  be 
zero,  to  making  ioo  +  ^>0  equal  to  631;;  if  v  be  3  .feet  per 
second,  p0  is  then  89  pounds  per  square  inch.  Since  these 
values  of  v  and  p  are  larger  than  the  usual  ones  for  a  city 
water  supply,  the  customary  practice  is  on  the  safe  side  for 
this  case,  but  for  the  high  velocities  and  pressures  used  for 
conveying  water  to  some  power  plants  it  would  not  give 
sufficient  security.  When  a  wave  of  dynamic  pressure 
travels  in  a  pipe  toward  a  closed  end,  the  water  hammer 
at  that  end  may  be  two  or  three  times  as  great  as  the  maxi- 
mum given  by  the  formula.  Air  chambers  at  the  ends  of 
pipes  slightly  reduce  the  effects  of  Water  hammer. 

Prob.  148a.  The  pressure-head  at  the  entrance  to  a  nozzle 
is  64.0  feet  when  there  is  no  flow  and  22.8  feet  when  the  water 
is  flowing.  The  pipe  is  1500  feet  long  and  4  inches  in  diameter, 


ART.  149  MOVING   VANES  395 

and  the  velocity  in  the  pipe  is  4.2  feet  per  second  when  the  valve 
at  the  nozzle  entrance  is  open.  Compute  the  excess  dynamic 
unit  pressure  when  the  valve  is  closed  in  one  second,  and  also 
when  it  is  closed  in  0.4  seconds. 

Prob.  1486.  For  the  data  of  the  last  problem,  find  the  pres- 
sures due  to  water  hammer  for  the  two  cases  at  a  point  in  the 
pipe  which  is  at  a  distance  x  from  the  valve  (see  Fig.  1486),  if 
the  elevation  of  this  point  below  the  water  level  of  the  reservoir 
is  23  feet. 

ART.  149.     MOVING  VANES 

A  vane  is  a  plane  or  curved  surface  which  moves  in  a 
given  direction  under  the  dynamic  pressure  exerted  by  an 
impinging  jet  or  stream.  The  direction  of  the  motion  of 
the  vane  depends  upon  the  conditions  of  its  construction; 
for  example,  the  vanes  of  a  water  wheel  can  only  move  in  a 
circumference  around  its  axis.  The  simplest  case  for  con- 
sideration, however,  is  that  where  the  motion  is  in  a  straight 
line,  and  this  alone  will  be  considered  in  this  article.  The 
plane  of  the  stream  and  vane  is  to  be  taken  as  horizontal, 
so  that  no  direct  action  of  gravity  can  influence  the  action 
of  the  jet. 

Let  a  jet  with  the  velocity  v  impinge  upon  a  vane  which 
moves  in  the  same  direction  with  the  velocity  u,  and  let  the 
velocity  of  the  jet  relative  to  the  sur- 
face at  the  point  of  exit  make  an  angle 
/?  with  the  reverse  direction  of  u,  as 
shown  in  Fig.  149a.  The  velocity  of 
the  stream  relative  to  the  surface  is 

v—u.  and  the  dynamic  pressure  is  the 

.-    1  >  FIG.  149a 

same  as  if  the  surface  were  at  rest  and 

the  stream  moving  with  the  absolute  velocity  v—u.  Hence 
formula  (145)x  directly  applies,  replacing  v  by  v—u  and  0 
by  180°—  /?,  and  the  dynamic  pressure  is 


o 


396  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

In  this  formula  W  is  not  the  weight  of  the  water  which 
issues  from  the  nozzle,  but  that  which  strikes  and  leaves 
the  vane,  or  W=wa(v—u)\  for  under  the  condition  here 
supposed  the  vane  moves  continually  away  from  the  nozzle, 
and  hence  does  not  receive  all  the  water  which  it  delivers. 

Another  method  of  deducing  the  last  equation  is  as  fol- 
lows: At  the  point  of  exit  let  lines  be  drawn  representing 
the  velocities  v—  u  and  u\  then,  completing  the  parallelo- 
gram, the  line  vl  is  the  absolute  velocity  of  the  departing 
jet  (Art.  30).  Let  6  be  the  angle  which  vl  makes  with  the 
direction  of  u,  and  /?  as  before  the  angle  between  v  —  u  and 
the  reverse  direction  of  u.  Then  the  dynamic  pressure  on 
the  vane  is  that  due  to  the  absolute  impulse  of  the  entering 
and  departing  streams:  the  former  of  these  is  W  .v/g  and 
the  latter  is  W  .  vt  co$6/g.  Hence  the  resultant  dynamic 
pressure  in  the  direction  of  the  motion  of  the  vane  is  the 
difference  of  these  impulses,  or 


But  from  the  triangle  between  v1  and  u 

vl  cos/9  =u  —  (v—u)  cos/? 
Inserting  this,  the  value  of  the  dynamic  pressure  is 


& 

which  is  the  same  as  that  found  before.  If  /?  =  180°  there  is 
no  pressure,  and  if  /?  =  0°  the  stream  is  completely  reversed, 
and  P  attains  its  maximum  value.  The  dynamic  pressure 
may  be  exerted  with  different  intensities  upon  different 
parts  of  the  vane,  but  its  total  value  in  the  direction  of  the 
motion  is  that  given  by  the  formula. 

Usually  the  direction  of  the  motion  is  not  the  same  as 
that  of  the  jet.     This  case  is  shown  in  Fig.  1496,  where  the 


ART.  149 


MOVING  VANES 


397 


FIG.  1496 


arrow  marked  F  designates  the  direction  of  the  impinging 
jet,  and  that  marked  P  the  direction  of  the  motion  of  the 
vane,  a  being  the  angle  be- 
tween them.  The  jet  hav- 
ing the  velocity  v  impinges 
upon  the  vane  at  A,  and  in 
passing  over  it  exerts  a 
dynamic  pressure  P  which 
causes  it  to  move  with  the 
velocity  u.  At  A  let  lines 
be  drawn  representing  the 
intensities  and  directions  of  v  and  M,  and  let  the  parallelo- 
gram of  velocities  be  formed  as  shown  ;  the  line  V  then  rep- 
resents the  velocity  of  the  water  relative  to  the  vane  at  A. 
The  stream  passes  over  the  surface  and  leaves  it  at  B  with 
the  same  relative  velocity  V,  if  not  retarded  by  friction  or 
foam.  At  B  let  lines  be  drawn  to  represent  u  and  V,  and 
let  /?  be  the  angle  which  V  makes  with  the  reverse  direction 
of  u  ;  let  the  parallelogram  be  completed,  giving  vl  for  the 
absolute  velocity  of  the  departing  water,  and  let  0  be  the 
angle  which  it  makes  with  u.  The  total  pressure  in  the 
direction  of  the  motion  is  now  to  be  regarded  as  that  caused 
by  the  components  in  that  direction  of  the  initial  and  the 
final  impulse  of  the  water.  The  impulse  of  the  stream 
before  striking  the  vane  is  W  .  v/g  and  its  component  in  the 
direction  of  the  motion  is  W  .v  cosa/g.  The  impulse  of  the 
stream  as  it  leaves  the  vane  is  W  .vjg  and  its  component 
in  the  direction  of  the  motion  is  W  .vt  cosO/g.  The  differ- 
ence of  these  components  is  the  resultant  dynamic  pressure 
in  the  given  direction,  or 


o 

This  is  a  general  formula  for  the  dynamic  pressure  in  any 
given  direction  upon  a  vane  moving  in  a  straight  line,  if 
a  and  6  be  the  angles  between  that  direction  and  those  of 


398  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xii 

v  and  vlt  If  the  surface  be  at  rest  v  and  v1  are  equal  and 
the  formula  reduces  to  (145)2. 

If  it  be  required  to  find  the  numerical  value  of  P,  the 
given  data  are  the  velocities  v  and  u  and  the  angles  or 
and  ft.  The  term  vt  cosd  is  hence  to  be  expressed  in  terms 
of  these  quantities.  From  the  triangle  at  B  between  vl 
and  u,  there  is  found 

v±  cos#  =u  —  V  cos/? 

und  substituting  this  the  formula  becomes 

_     v  cosa  —u+V  cos/? 

g 

which  is  often  a  more  convenient  form  for  discussion. 
The  value  of  V  is  found  from  the  triangle  at  A  between 
u  and  v,  thus: 

V 2  =  u2  +  v2  —  2uv  cosa 

and  hence  the  dynamic  pressure  P  is  fully  determined  in 
terms  of  the  given  data. 

In  order  that  the  stream  may  enter  tangentially  upon 
the  vane,  and  thus  prevent  foam,  the  curve  of  the  vane  at  A 
should  be  tangent  to  the  direction  of  V.  This  direction 
can  be  found  by  expressing  the  angle  <j>  in  terms  of  the 
given  angle  a.  Thus  from  the  relation  between  the  sides 
and  angles  of  the  triangle  included  between  u,  v,  and  V 
there  is  found 

sin  (<j>  —  a) /sin <j>  =  u/v 

which  is  easily  reduced  to  the  form 

u 

cot<£  =cota : — • 

v  sine* 

from  which  <£  can  be  computed  when  u,  v,  and  a  are  given. 
For  example,  if  u  be  equal  to  %v,  and  if  a  be  30°,  then 
cot  $  is  0.732,  whence  the  angle  <£  should  be  53'!°  in  order 
that  the  jet  may  enter  without  impact.  If  the  angle  made 


ART.  150  WORK    FROM   MOVING   VANES  399 

by  the  vane  with  the  direction  of  motion  be  greater  or 
less  than  this  value  some  loss  due  to  impact  will  result  at 
the  given  speed. 

Prob.  149a.  Given  ^  =  70.7  and  v=ioo.o  feet  per  second, 
a  =  45°  and  /?  =  3O°.  Compute  the  dynamic  pressure  P  when 
the  quantity  of  water  striking  the  vane  is  0.6  cubic  feet  per 
second. 

Prob.  1496.  Given  w  =  86.6  and  ?;=ioo.o  feet  per  second, 
and  a  =  30°.  What  should  be  the  value  of  the  angle  <fr  in  order 
that  no  loss  by  impact  may  occur?  Draw  the  parallelogram 
showing  the  velocities  u,  v,  and  V. 

ART.  150.     WORK  DERIVED  FROM  MOVING  VANES 

The  work  imparted  to  a  moving  vane  by  the  energy  of 
the  impinging  water  is  equal  to  the  product  of  the  dynamic 
pressure  P,  which  is  exerted  in  the  direction  of  the  motion 
and  the  space  through  which  it  moves.  If  u  be  the  space 
described  in  one  second,  or  the  velocity  of  the  vane,  the 
work  per  second  is 

k=Pu 

This  expression  is  now  to  be  discussed  in  order  to  deter- 
mine the  value  of  u  which  makes  k  a  maximum. 

When  the  vane  moves  in  a  straight  line  in  the  same 
direction  as  the  impinging  jet  and  the  water  enters  it 
tangentially,  as  shown  in  Fig.  1456,  the  work  imparted 
is  found  by  inserting  for  P  its  value  from  (145)j.  If  a  be 
the  area  of  the  cross-section  of  the  jet  and  w  the  weight 
of  a  cubic  unit  of  water,  the  weight  W  is  wa(v  —u),  and  then 


5 

The  value  of  u  which  renders  k  a  maximum  is  obtained  by 
equating  to  zero  the  derivative  of  k  with  respect  to  u,  or 

dk  ^wa, 

-^  =  (i+cosp)~(v2 


400  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

from  which  the  value  of  u  is  Jv,  and  accordingly 

v3 
k  =¥8T(i  +cos/?)wa— 

is  the  maximum  work  that  can  be  done  by  the  vane  in  one 
second.  The  theoretic  energy  of  the  impinging  jet  is 

v2          v3 
K  =  W—=wa- 

2g  2g 

and  the  efficiency  of  the  vane  is  the  ratio  of  the  effective 
work  of  the  vane  to  the  theoretic  energy  of  the  water,  or 

e  =  k/K  =•&(!  +  cos/?) 

If  /?  =  1 80°,  the  jet  glides  along  the  vane  without  producing 
work  and  e  =o ;  if  /?  =90°,  the  water  departs  from  the  vane 
normal  to  its  original  direction  and  e=^r\  ^  /?=0°>  the 
direction  of  the  stream  is  reversed  and  e=ffi 

It  appears  from  the  above  that  the  greatest  efficiency 
which  can  be  obtained  by  a  vane  moving  in  a  straight  line 
under  the  impulse  of  a  jet  of  water  is  |f;  that  is,  the  effec- 
tive work  is  only  about  59  percent  of  the  theoretic  energy 
attainable.  This  is  due  to  two  causes:  first,  the  quantity 
of  water  which  reaches  and  leaves  the  vane  per  second  is 
less  than  that  furnished  by  the  nozzle  or  mouthpiece  from 
which  the  water  issues;  and,  secondly,  the  water  leaving 
the  vane  still  has  an  absolute  velocity  of  %v.  A  vane 
moving  in  a  straight  line  is  therefore  a  poor  arrangement 
for  utilizing  energy,  and  it  will  also  be  seen  upon  reflection 
that  it  would  be  impossible  to  construct  a  motor  in  which 
a  vane  would  move  continually  in  the  same  direction  away 
from  a  fixed  nozzle.  The  above  discussion  therefore  gives 
but  a  rude  approximation  to  the  results  obtainable  under 
practical  conditions.  It  shows  truly,  however,  that  the 
efficiency  of  a  jet  which  is  deflected  normally  from  its  path 
is  but  one-half  of  that  obtainable  when  a  complete  reversal 
of  direction  is  made. 


ART.  150  WORK   FROM   MOVING    VANES  401 

Water  wheels  which  act  under  the  impulse  of  a  jet 
consist  of  a  series  of  vanes  arranged  around  a  circum- 
ference which  by  the  motion  are  brought  in  succession 
before  the  jet.  In  this  case  the  quantity  of  water  which 
leaves  the  wheel  per  second  is  the  same  as  that  which  enters 
it,  so  that  W  does  not  depend  on  the  velocity  of  the  vanes, 
as  in  the  preceding  case,  but  is  a  constant  whose  value  is 
wq,  where  q  is  the  quantity  furnished  per  second.  A  close 
estimate  of  the  efficiency  of  a  series  of  such  vanes  can  be 
made  by  considering  a  single  vane  and  taking  W  as  a  con- 
stant. The  water  is  supposed  to  impinge  tangentially 
and  the  vane  to  move  in  the  same  direction  as  the  jet 
(Fig.  149a).  Then  the  work  imparted  per  second  by  the 
water  to  the  moving  vane  is 


o 

This  becomes  zero  when  u=o  or  when  u=v,  and  it  is  a 
maximum  when  u  =  %v,  or  when  the  vane  moves  with  one- 
half  the  velocity  of  the  jet.  Inserting  this  value  of  «, 


and,  dividing  this  by  the  theoretic  energy  of  the  jet,  the 
efficiency  of  the  vane  is  found  to  be 


When  /?  =  i8o°,  the  jet  merely  glides  along  the  surface 
without  performing  work  and  e=o',  when  ^=90°,  the  jet 
is  deflected  normally  to  the  direction  of  the  motion  and 
e  =  \  ;  when  /?  =  o°,  a  complete  reversal  of  direction  is  ob- 
tained and  the  efficiency  attains  its  maximum  value  e  =  i  . 

These  conclusions  apply  closely  to  the  vanes  of  a  water 
wheel  which  are  so  shaped  that  the  water  enters  upon 
them  tangentially  in  the  direction  of  the  motion.  If  the 
vanes  are  plane  radial  surfaces,  as  in  simple  paddle  wheels, 


402  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

the  water  passes  away  normally  to  the  circumference,  and 
the  highest  obtainable  efficiency  is  about  50  percent.  If 
the  vanes  are  curved  backward  the  efficiency  becomes 
greater,  and,  neglecting  losses  in  impact  and  friction,  it 
might  be  made  nearly  unity,  and  the  entire  energy  of  the 
stream  be  realized,  if  the  water  could  both  enter  and  leave 
the  vanes  in  a  direction  tangential  to  the  circumference. 
The  investigation  shows  that  this  is  due  to  the  fact  that  the 
water  leaves  the  vanes  without  velocity;  for,  as  the  ad- 
vantageous velocity  of  the  vane,  is  %v,  the  water  upon  its 
surface  has  the  relative  velocity  v—%v=%v;  then,  if  /?=o°, 
its  absolute  velocity  as  it  leaves  the  vane  is  %v  —  %v=o. 
If  the  velocity  of  the  vanes  is  less  or  greater  than  half  the 
velocity  of  the  jet,  the  efficiency  is  lessened,  although 
slight  variations  from  the  advantageous  velocity  do  not 
practically  influence  the  value  of  e. 

Prob.  150.  A  nozzle  0.125  feet  in  diameter,  whose  coeffi- 
cient of  discharge  is  0.95,  delivers  water  under  a  head  of  82  feet 
against  a  series  of  small  vanes  on  a  circumference  whose  diam- 
eter is  18.5  feet.  Find  the  most  advantageous  velocity  of  revo- 
lution of  the  circumference. 

ART.  151.     REVOLVING  VANES 

When  vanes  are  attached  to  an  axis  around  which  they 
move,  as  is  the  case  in  water  wheels,  the  dynamic  pressure 
which  is  effective  in  causing  the  motion  is  that  tangential 
to  the  circumferences  of  revolution ;  or  at  any  given  point 
this  effective  pressure  is  normal  to  a  radius  drawn  from  the 
point  to  the  axis.  In  Fig.  151  are  shown  two  cases  of  a 
rotating  vane;  in  the  first  the  water  passes  outward  or 
away  from  the  axis,  and  in  the  second  it  passes  inward  or 
toward  the  axis.  The  reasoning,  however,  is  general  and 
will  apply  to  both  cases.  At  A,  where  the  jet-  enters  upon 
the  vane,  let  v  be  its  absolute  velocity,  V  its  velocity  relative 
to  the  vane,  and  u  the  velocity  of  the  point  A  ;  draw  u  nor- 


ART.  151 


REVOLVING  VANES 


403 


mal  to  the  radius  r  and  construct  the  parallelogram  of  ve- 
locities as  shown,  a  being  the  angle  between  the  directions 
of  u  and  v,  and  <j>  that  between  u  and  V.  At  B,  where  the 
water  leaves  the  vane,  let  uv  be  the  velocity  of  that  point 
normal  to  the  radius  r1?  and  Vl  the  velocity  of  the  water 
relative  to  the  vane ;  then  constructing  the  parallelogram, 


U        i 


FIG.  151 

the  resultant  of  ut  and  Vt  is  vlt  the  absolute  velocity  of  the 
departing  water.  Let  /?  be  the  angle  between  Vl  and  the 
reverse  direction  of  ult  and  0  be  the  angle  between  the 
directions  of  v^  and  u^. 

The  total  dynamic  pressure  exerted  in  the  direction  of 
the  motion  will  depend  upon  the  values  of  the  impulse  of 
the  entering  and  departing  streams.  The  absolute  impulse 
of  the  water  before  entering  is  W  .  v/g,  and  that  of  the  water 
after  leaving  is  W  .vjg.  Let  the  components  of  these  in 
the  directions  of  the  motion  of  the  vane  at  entrance  and 
departure  be  designated  by  P  and  Pv  ;  then 

_ 


These,  however,  cannot  be  subtracted  to  give  the  resultant 
dynamic  pressure,  as  was  done  in  the  case  of  motion  in  a 
straight  line,  because  their  directions  are  not  parallel,  and 


404  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

the  velocities  of  their  points  of  application  are  not  equal. 
The  resultant  dynamic  pressure  is  not  important  in  cases 
of  this  kind,  but  the  above  values  will  prove  useful  in  the 
next  article  in  investigating  the  work  that  can  be  delivered 
by  the  vane. 

If  n  be  the  number  of  revolutions  around  the  axis  in  one 
second,  the  velocities  u  and  %  are 

u  =  27irn         u^  =  2nr1n 
and  accordingly  the  relation  obtains 

u1/u=r1/r         or         ^1r=^r1 

which  shows  that  the  velocities  of  the  points  of  entrance 
and  exit  are  directly  proportional  to  their  distances  from 
the  axis.  If  r  and  rx  are  infinity,  u  equals  u^  and  the  case 
is  that  of  motion  in  a  straight  line  as  discussed  in  Art.  149. 

The  relative  velocities  Vl  and  V  are  connected  with  the 
velocities  of  rotation  %  and  u  by  a  simple  relation.  To  de- 
duce it,  imagine  an  observer  standing  on  the  outward-flow 
vane  and  moving  with  it;  he  sees  a  particle  of  weight  w 
at  A  which  to  him  appears  to  have  the  velocity  V,  while  the 
same  particle  at  B  appears  to  have  the  velocity  V1't  the 
difference  of  their  kinetic  energies  or  w(V*  —  V2)/2g  is 
the  apparent  gain  of  the  wheel-energy.  Again,  consider 
an  observer  standing  on  the  earth  and  looking  down 
upon  the  vane ;  from  his  point  of  view  the  energy  gained  is 
w(u*  —  u2)/2g.  Now  these  two  expressions  for  the  gain  of 
the  wheel  in  energy  must  be  equal,  or 

VS-V'-its-u*  (151) 

and  this  is  the  formula  by  which  Vl  is  to  be  computed  when 
V  and  the  velocities  of  rotation  are  known.  The  same 
reasoning  applies  to  the  inward-flow  vane  by  using  the 
word  loss  instead  of  gain,  and  the  same  formula  results. 

The  given  data  for  a  revolving  vane  are  the  angles  a, 


ART.  152  WORK   FROM   REVOLVING   VANES  405 

0,  and  /?,  the  radii  r  and  rlt  the  velocity  v,  the  number  of 
revolutions  per  second,  and  the  weight  of  water  delivered 
to  the  vane  per  second.  The  value  of  v  cos  a,  and  hence 
that  of  P,  is  immediately  known.  From  the  speed  of  revo- 
lution the  velocities  u  and  u^  are  found.  The  relative 
velocity  V  is,  from  the  triangle  between  u  and  v, 

V  =v  sin  a/sin  <j> 
and  by  (151)  the  relative  velocity  Vt  is  then  found  from 

V12=U12-M2+V2 

Lastly,  the  value  of  v^  cos#  is,  from  the  triangle  between 
u±  and  Vlt 

vl  cosO  =  ul  —  Vl  cos/? 

and  accordingly  the  values  of  P  and  Pl  are  fully  deter- 
mined. Numerical  values  of  these,  however,  are  never 
needed,  but  the  work  due  to  them  is  to  be  found,  as  will 
be  explained  in  the  next  article. 

Prob.  151a.  If  a  point  14  inches  from  the  axis  moves  with  a 
uniform  velocity  of  62  feet  per  second,  how  many  revolutions 
does  it  make  per  minute? 

Prob.  1516.  Given  r=2  feet,  ^  =  3  feet,  a  =45°,  ^  =  90°, 
v=  100  feet  per  second,  and-w  =  6  revolutions  per  second.  Com- 
pute the  velocities  u,  ult  V,  and  Vr 

ART.  152.     WORK  DERIVED  FROM  REVOLVING  VANES 

The  investigation  in  Art.  150  on  the  work  and  efficiency 
of  a  revolving  vane  supposes  that  all  its  points  move  with 
the  same  velocity,  and  that  the  water  enters  upon  it  in  the 
same  direction  as  that  of  its  motion,  or  that  a=o.  This 
cannot  in  general  be  the  case  in  water  motors,  as  then  the 
jet  would  be  tangential  to  the  circumference  and  no  water 
could  enter.  To  consider  the  subject  further  the  reasoning 
of  the  last  article  will  be  continued,  and,  using  the  same 
notation,  it  will  be  plain  that  the  work  of  a  series  of  vanes 
arranged  around  a  wheel  may  be  regarded  as  that  due 


406  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

to  the  impulse  of  the  entering  stream  in  the  direction  of  the 
motion  around  the  axis  minus  that  due  to  the  impulse  of 
the  departing  stream  in  the  same  direction,  or 

k=Pu-P1u1 

Here  P  and  P±  are  the  pressures  due  to  the  impulse  at  A 
and  B  (Fig.  151),  and  inserting  their  values  as  found, 


This  is  a  general  formula  applicable  to  the  work  of  all  wheels 
of  outward  or  inward  flow,  and  it  is  seen  that  the  useful 
work  k  consists  of  two  parts,  one  due  to  the  entering  and 
the  other  to  the  departing  stream. 

Another  general  expression  for  the  work  of  a  series  of 
vanes  may  be  established  as  follows:  Let  v  and  vl  be  the 
absolute  velocities  of  the  entering  and  departing  water; 
the  theoretic  energy  of  this  water  is  W.v*/2g,  and  when  it 
leaves  the  wheel  it  still  has  the  energy  W.vl2/2g.  Neglect- 
ing losses  of  energy  in  impact  and  friction  the  work  that 
can  be  derived  from  the  wheel  is 

7;2  _  ni   2 

k  =  W~~  (152), 

This  is  a  formula  of  equal  generality  with  the  preceding, 
and  like  it  is  applicable  to  all  cases  of  the  conversion  of 
energy  into  work  by  means  of  impulse  or  reaction.  In  both 
formulas,  however,  the  plane  of  the  vane  is  supposed  to  be 
horizontal,  Nso  that  no  fall  occurs  between  the  points  of  en- 
trance and  exit. 

Formula  (151)  may  be  demonstrated  in  another  way 
by  equating  the  values  of  k  in  the  preceding  formulas  ;  thus 

uv  cosa  —u^  cos#  =%(v2  —v^) 
Now  from  the  triangle  at  A  between  u  and  v 
i)*  =  V2  —  n2  +  2uv  cosa 


ART.  152 


WORK  FROM  REVOLVING  VANES 


407 


and  from  the  triangle  at  B  between  %  and  v^ 

vi2  =  VV  —  US  +  2U1V1  COS0 

Inserting  these  values  of  v2  and  v^2  the  equation  reduces  to 
Vl2-V2=ul2-u2 

This  shows  that  if  ut  be  greater  than  u,  as  in  the  outward- 
flow  vane  of  the  first  diagram  of  Fig.  151,  then  Vt  is  greater 
than  V\  ii  HI  is  less  than  u,  as  in  an  inward-flow  vane, 
then  Vl  is  less  than  V. 

The  above  principles  will  now  be  applied  to  the  simple 
case  of  an  outward-flow  wheel  driven  by  a  fixed  nozzle,  as 
in  Fig.  152a.  The  wheel  is  so  built  that  r  =  2  feet,  rt  =  3  feet, 
a  =45°,  0=90°,  and  ^  =  30°.  The  velocity  of  the  water 


FIG.  152a  FIG.  1526 

issuing  from  the  nozzle  is  v  =  ioofeet  per  second,  and  the 
discharge  per  second  is  2.2  cubic  feet.  It  is  required  to 
find  the  work  of  the  wheel  and  the  efficiency  when  its  speed 
i§  337-5  revolutions  per  minute. 

The  theoretic  work  of  the  stream  per  second  is  the 
weight  delivered  per  second  multiplied  by  its  velocity- 
head,  or 

k  =62.5  X 2. 2  Xo. 01555  Xioo2  =  2i  380  foot-pounds 

which  gives  38.9  theoretic  horse-powers.  The  actual 
work  of  the  wheel,  neglecting  losses  in  foam  and  friction, 
can  be  computed  either  from  (152)!  or  (152)2.  In  order 
to  use  the  first  of  these,  however,  the  velocities  u,  ult  vlt 
and  the  angle  6  must  be  found,  and  to  use  the  second,  v± 


408  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

must  be  found;   in  each  case  V  and  Fx  must  be  determined. 

The  velocities  u  and  u^  are  found  from  the  given  speed 

of  5.625  revolutions  per  second,  thus: 

u  =2    X3.i4i6  X2  X5.625  =   70.71  feet  per  second 
^  =  1^X70.71  =106.06  feet  per  second 

The  relative  velocity  V  at  the  point  of  entrance  is  found 

from  the  triangle  between  V  and  v,  which  in  this  case  is 

right-angled;  thus 

V  =v  cos(<£  —  a)  =i'cos  45°  =  70. 7 1  feet  per  second 

The  relative  velocity  Vl  at  the  point  of  exit  is  found  from 
the  relation  (151),  which  gives  1^=^  =  106.06  feet  per 
second.  And  since  %  and  Vl  are  equal,  v±  bisects  the  angle 
between  V^  and  ult  and  accordingly 

0=i(i8o°-/?)=  75  degrees 
The  value  of  the  absolute  velocity  vl  then  is 

vl  =  2%cos#  =  54.90  feet  per  second 
and  v^/2g  is  the  velocity-head  lost  in  the  escaping  water. 

The  work  of  the  wheel  per  second,  computed  either 
from  (152)!  or  (152)2,  is  now  found  to  be  £  =  14934  foot- 
pounds or  27.2  horse-powers,  and  hence  the  efficiency, 
or  the  ratio  of  this  work  to  the  theoretic  work,  is  e  =0.699. 
Thus  30.1  percent  of  the  energy  of  the  water  is  lost,  owing 
to  the  fact  that  the  water  leaves  the  wheel  with  such  a 
large  absolute  velocity. 

In  this  example  the  speed  given,  337.5  revolutions  per 
minute,  is  such  that  the  direction  of  the  relative  velocity 
V  is  tangent  to  the  vane  at  the  point  of  entrance.  For 
any  other  speed  this  will  not  be  the  case,  and  thus  work 
will  be  lost  in  shock  and  foam.  It  is  observed  also  that 
the  approach  angle  a  is  one-half  of  the  entrance  angle  <£ ; 
with  this  arrangement  the  velocities  u  and  V  are  equal, 
as  also  Ut  and  V^  Had  the  angle  /?  been  made  smaller  the 
efficiency  would  have  been  higher. 


ART.  153 


REVOLVING  TUBES 


409 


Prob.  152a.  Compute  the  power  and  efficiency  for  the  above 
example  if  the  angle  ft  be  15°  instead  of  30°.  Explain  why  ft 
cannot  be  made  very  small. 

Prob.  1526.  Compute  the  power  and  efficiency  of  the  inward 
flow  wheel  in  Fig.  1526,  when  r  =  3  feet,  rl  =  2  feet,  a  =  30°,  <f>  =  60°, 
/?  =  6o°,  z;=ioo  feet  per  second,  q  =  2.2  cubic  feet  per  second, 
and  the  speed  being  184  revolutions  per  minute. 

ART.  153.     REVOLVING  TUBES 

The  water  which  glides  over  a  vane  can  never  be  under 
static  pressure,  but  when  two  vanes  are  placed  near  together 
and  connected  so  as  to  form  a  closed  tube,  there  may  exist 
in  it  static  pressure  if  the  tube  is  filled.  This  is  the  condi- 
tion in  turbine  wheels,  where  a  number  of  such  tubes,  or 
buckets,  are  placed  around  an  axis  and  water  is  forced 
through  them  by  the  static  pressure  of  a  head.  The  work 
in  this  case  is  done  by  the  dynamic  pressure  exactly  as  in 
vanes,  but  the  existence  of  the  static  pressure  renders  the 
investigation  more  difficult. 

The  simplest  instance  of  a  revolving  tube  is  that  of  an 
arm  attached  to  a  vessel  rotating  about  a  vertical  axis, 
as  in  Fig.  153.  It  was  shown  in  Art.  31 
that  the  water  surface  in  this  case  as- 
sumes the  form  of  a  paraboloid,  and 
if  no  discharge  occurs  it  is  clear  that 
the  static  pressures  at  any  two  points 
B  and  A  are  measured  by  the  pressure- 
heads  //!  and  H  reckoned  upwards  to 
the  parabolic  curve,  and,  if  the  veloci- 
ties of  those  points  are  u^  and  u,  that 


FIG.  153 


Now  suppose  an  .orifice  to  be  opened 
in  the  end  of  the  tube  and  the  flow  to 
occur  while  at  the  same  time  the  revolution  is  continued. 


410  DYNAMIC  PRESSURE  OF  WATER  CHAP,  xn 

The  velocities  Vl  and  V  diminish  the  pressure-heads  so 
that  the  piezometric  line  is  no  longer  the  parabola  but 
some  curve  represented  by  the  lower  broken  line  in  the 
figure.  Then,  according  to  the  theorem  of  Art.  32,  that 
pressure -head  plus  velocity-head  remains  constant  during 
steady  flow,  if  no  loss  of  energy  occurs, 

V,2     u2  V2     u2 

H+lL  _!!!_«#+-!:       !»£  (153) 

2g         2g  2g          2g 

in  which  Hl  and  H  are  the  heads  due  to  the  actual  static 
pressures.  This  is  the  theorem  which  gives  the  relation 
between  pressure-head,  velocity-head,  and  rotation-head 
at  any  point  of  a  revolving  tube  or  bucket.  If  the  tube 
is  only  partly  full,  so  that  the  flow  occurs  along  one  side, 
like  that  of  a  stream  upon  a  vane,  then  there  is  no  static 
pressure,  and  the  formula  becomes  the  same  as  (151). 

An  apparatus  like  Fig.  153,  but  having  a  number  of 
arms  from  which  the  flow  issues,  is  called  a  reaction  wheel, 
since  the  dynamic  pressure  which  causes  the  revolution  is 
wholly  due  to  the  reaction  of  the  issuing  water.  To  in- 
vestigate it,  the  general  formula  (152)  x  may  be  used.  Mak- 
ing u  =  o,  the  work  done  upon  the  wheel  by  the  water  is 
k_w  -UM  cosfl  =w^Vi  cos/? -u,2 

g  g 

But  since  there  is  no  static  pressure  at  the  point  B,  the 
value  of  Vl  is,  from  (153),  or  also  from  Art.  31, 


The  work  that  can  be  derived  from  the  wheel  now  is 


g 
This  becomes  nothing  when  ut  =o,  or  when  u^  =  2gh  cot2/?, 

and  by  equating  the  first  derivative  to  zero  it  is  found  that 
k  becomes  a  maximum  when  the  velocity  is  given  by 

M*-^L.-ek 
Ml  "sin/?    g"      , 


ART.  153  REVOLVING    TUBES  41  j 

Inserting  this  advantageous  velocity,  the  maximum  work  is 

k=Wh(i-  sin/?) 

and  therefore  the  efficiency  of  the  reaction  wheel  is 

e  =  i  -  sin/? 

When  /?  =  9o°,  both  ul  and  e  become  o,  for  then  the  direc- 
tion of  the  stream  is  normal  to  the  circumference  and  no 
reaction  can  occur  in  the  direction  of  revolution.  When 
/?  =  o  the  efficiency  becomes  unity,  but  .the  velocity  u^ 
becomes  infinity.  In  the  reaction  wheel,  therefore,  high 
efficiency  can  only  be  secured  by  making  the  direction  of 
the  issuing  water  directly  opposite  to  that  of  the  revolu- 
tion, and  by  having  the  speed  very  great.  If  ft  =  19°,$ 
or  sin/?=J,  the  advantageous  velocity  ul  becomes  \/2gh 
and  e  becomes  0.67.  The  effect  of  friction  of  the  water 
on  the  sides  of  the  revolving  tube  is  not  here  considered, 
but  this  will  be  done  in  Art.  163. 

Prob.  153a.  Compute  the  theoretic  efficiency  of  the  reaction 
wheel  when  #  =  180°,  /?  =  o°,  and  ul  =  V/2gh. 

Prob.  1536.  A  reaction  wheel  has  /3  =  3o°,  ^  =  0.302  meters, 
and  ^  =  4.5  meters.  Compute  the  most  advantageous  number 
of  revolutions  per  minute.  If  the  quantity  of  water  delivered 
to  the  wheel  is  1600  liters  per  minute,  compute  the  power  of  the 
wheel  in  metric  horse-powers  and  in  kilowatts. 

Prob.  153^.  When  /  is  in  meters,  v  in  meters  per  second,  and 
p,  pi,  and  pQ  are  in  kilograms  per  square  centimeter,  the  formulas 
(148)3  for  water  hammer  become 


the  first  of  which  is  to  be  used  when  t  is  greater  than  0.0014042 
and  the  second  when  t  is  equal  to  or  less  than  it,  /  being  in 
meters. 


412  WATER  WHEELS  CHAP,  xin 


CHAPTER  XIII 
WATER    WHEELS 

ART.  154.     CONDITIONS  OF.  HIGH  EFFICIENCY 

A  hydraulic  motor  is  an  apparatus  for  utilizing  the  energy 
of  a  waterfall.  It  generally  consists  of  a  wheel  which  is 
caused  to  revolve  either  by  the  weight  of  water  falling  from 
a  higher  to  a  lower  level,  or  by  the  dynamic  pressure  due  to 
the  change  in  direction  and  velocity  of  a  moving  stream. 
When  the  water  enters  at  only  one  part  of  the  circumfer- 
ence the  apparatus  is  called  a  water-wheel;  when  it  enters 
around  the  entire  circumference  it  is  called  a  turbine.  In 
this  chapter  and  the  next  these  two  classes  of  motors  will 
be  discussed  in  order  to  determine  the  conditions  which 
render  them  most  efficient.  Overshot  wheels,  which  move 
under  the  weight  of  water  caught  in  their  buckets,  and  under- 
shot wheels,  which  move  under  the  impact  of  a  flowing 
stream,  are  forms  that  have  been  used  for  many  centuries. 
Impulse  wheels,  which  owe  their  motion  to  a  jet  of  water 
striking  their  vanes  with  high  velocity  were  perfected  in 
the  eighteenth  century. 

The  efficiency  e  of  a  motor  ought,  if  possible,  to  be  inde- 
pendent of  the  amount  of  water  used,  or  if  not,  it  should  be 
the  greatest  when  the  water  supply  is  low.  This  is  very 
difficult  to  attain.  It  should  be  noted,  however,  that  it 
is  not  the  mere  variation  in  the  quantity  of  water  which 
causes  the  efficiency  to  vary,  but  it  is  the  losses  of  head 
which  are  consequent  thereon.  For  instance,  when  water 
is  low,  gates  must  be  lowered  to  diminish  the  area  of  ori- 


ART.  154  CONDITIONS    OF   HlGH   EFFICIENCY  413 

fices,  and  this  produces  sudden  changes  of  section  which 
dimmish  the  effective  head  h.  A  complete  theoretic  ex- 
pression for  the  efficiency  will  hence  not  include  Wt  the 
weight  of  water  supplied  per  second,  but  it  should,  if  possi- 
ble, include  the  losses  of  energy  or  head  which  result  when 
W  varies.  The  actual  efficiency  of  a  motor  can  only  be 
determined  by  tests  with  the  friction  brake  (Art.  140) ; 
the  theoretic  efficiency,  as  deduced  from  formulas  like  those 
of  the  last  chapter,  will  as  a  rule  be  higher  than  the  actual, 
because  it  is  impossible  to  formulate  accurately  all  the 
sources  of  loss.  Nevertheless  the  deduction  and  discus- 
sion of  formulas  for  theoretic  efficiency  is  very  important 
for  the  correct  understanding  and  successful  construction 
of  hydraulic  motors. 

When  a  weight  of  water  W  falls  in  each  second  through 
the  height  h,  or  when  it  is  delivered  with  the  velocity  v, 
its  theoretic  energy  per  second  is 

K  =  Wh  or  K  =  W— 

2g 

The  actual  work  per  second  equals  the  theoretic  energy 
minus  all  the  losses  of  energy.  These  losses  may  be  divided 
into  two  classes:  first,  those  caused  by  the  transformation 
of  energy  into  heat ;  and  second,  those  due  to  the  velocity 
^  with  which  the  water  reaches  the  level  of  the  tail  race. 
The  first  class  includes  losses  in  friction,  losses  in  foam  and 
eddies  consequent  upon  sudden  changes  in  cross-section 
or  from  allowing  the  entering  water  to  dash  improperly 
against  surfaces;  let  the  loss  of  work  due  to  this  be  Wh't 
in  which  h'  is  the  head  lost  by  these  causes.  The  second 
loss  is  due  merely  to  the  fact  that  the  departing  water 
carries  away  the  energy  W.vl2/2g.  The  work  per  second 
imparted  by  the  water  to  the  wheel  then  is 


414  WATER  WHEELS  CHAP,  xnr 

and  dividing  this  by  the  theoretic  energy  the  efficiency  is 


in  which  v  is  the  velocity  due  to  the  head  h.  This  formula, 
although  very  general,  must  be  the  basis  of  all  discussions 
on  the  theory  of  water  wheels  and  motors.  .  It  shows  that 
e  can  only  become  unity  when  hr  =o  and  v±  =o,  and  accord- 
ingly the  two  following  fundamental  conditions  must  be 
fulfilled  in  order  to  secure  high  efficiency  : 

i  .  The  water  must  enter  and  pass  through  the  wheel 
without  losing  energy  in  friction  and  foam. 

2.  The  water  must  reach  the  level  of  the  tail  race 
without  absolute  velocity. 

These  two  requirements  are  expressed  in  popular  language 
by  the  well-known  maxim  '  '  the  water  should  enter  the 
wheel  without  shock  and  leave  without  velocity.  '  '  Here 
the  word  shock  means  that  method  of  introducing  the  water 
which  produces  foam  and  eddies. 

The  friction  of  the  wheel  upon  its  bearings  is  included  in 
the  lost  work  when  the  power  and  efficiency  are  actually 
measured  as  described  in  Art.  140.  But  as  this  is  not  a 
hydraulic  loss  it  should  not  be  included  in  the  lost  work  kr 
when  discussing  the  wheel  merely  as  a  user  of  water,  as  will 
be  done  in  this  chapter.  The  amount  lost  in  shaft  &nd  jour- 
nal friction  in  good  constructions  may  be  estimated  at  2 
or  3  percent  of  the  theoretic  energy,  so  that  in  discussing 
the  hydraulic  losses  the  maximum  value  of  e  will  not  be 
unity,  but  about  0.98  or  0.97.  This  will  usually  be  ren- 
dered considerably  smaller  by  the  friction  of  the  wheel 
upon  the  air  or  water  in  which  it  moves,  and  which  will 
here  not  be  regarded.  The  efficiency  given  by  (154)  is 
called  the  hydraulic  efficiency  to  distinguish  it  from  the 
actual  efficiency  as  determined  by  the  friction  brake. 


ART.  155 


OVERSHOT  WHEELS 


415 


Prob.  154.  A  wheel  using  70  cubic  feet  of  water  per  minute 
under  a  head  of  12.4  feet  has  an  efficiency  of  63  percent.  What 
effective  horse-power  does  it  deliver? 


ART.  155.     OVERSHOT  WHEELS 

In  the  overshot  wheel  the  water  acts  largely  by  its 
weight.  Fig.  155  shows  an  end  view  or  vertical  section, 
which  so  fully  illustrates  its  action  that  no  detailed  explan- 
ation is  necessary.  The  total  fall  from  the  surface  of  the 
water  in  the  head  race  or 
flume  to  the  surface  in  the 
tail  race  is  called  h,  and  the 
weight  of  water  delivered 
per  second  to  the  wheel  is 
called  W.  Then  the  theo- 
retic energy  per  second  im- 
parted to  the  wheel  is  Wh. 
It  is  required  to  determine 
the  conditions  which  will 
render  the  effective  work 
of  the  wheel  as  near  to  Wh  s± 
as  possible. 

The  total  fall  may  be  FIG.  155 

divided  into  three  parts ;  that  in  which  the  water  is  filling 
the  buckets,  that  in  which  the  water  is  descending  in  the 
filled  buckets,  and  that  which  remains  after  the  buckets 
are  emptied.  Let  the  first  of  these  parts  be  called  h0,  and 
the  last  hlt  In  falling  the  distance  h0  the  water  acquires  a 
velocity  VQ  which  is  approximately  equal  to  V  2ghQ,  and  then, 
striking  the  buckets,  this  is  reduced  to  u,  the  tangential 
velocity  of  the  wheel,  whereby  a  loss  of  energy  in  impact 
occurs.  It  then  descends  through  the  distance  h-h0-hlt 
acting  by  its  weight  alone,  and  finally,  dropping  out  of  the 
buckets,  reaches  the  level  of  the  tail  race  with  a  velocity 
which  causes  a  second  loss  of  energy.  Let  h'  be  the  head 


416  WATER  WHEELS  CHAP,  xm 

lost  in  entering  the  buckets,  and  let  vl  be  the  velocity  of 
the  water  as  it  reaches  the  level  of  the  tail  race.  Then  the 
hydraulic  efficiency  of  the  wheel  is  given  by  the  general 
formula  (154),  or 


h 

and  to  apply  it,  the  values  of  h'  and  v^  are  to  be  found.     In 
this  equation  v  is  the  velocity  due  to  the  head  h,orv=V-2gh. 

The  head  lost  in  impact  when  a  stream  of  water  with  the 
velocity  VQ  is  enlarged  in  section  so  as  to  have  the  smaller 
velocity  u,  is,  as  proved  in  Art.  74, 


7  '  ^ 


The  velocity  vl  with  which  the  water  reaches  the  tail  race 
depends  upon  the  velocity  u  and  the  height  hlm  Its  kinetic 
energy  as  it  leaves  the  buckets  is  W  .u2/2g,  the  potential 
energy  of  the  fall  h^  is  Whlt  and  the  resultant  kinetic  energy 
as  it  reaches  the  tail  race  is  W  .  v^/2g  ;  hence  the  value  of  vt  is 


Inserting  these  values  of  h'  and  vl  in  the  formula  for  e,  and 
placing  for  v2  its  equivalent  2gh,  there  is  found 


2U 

e  =  i 


-   2gh 

The  value  of  u  which  renders  e  a  maximum  is  found  by 
equating  the  first  derivative  to  zero,  which  gives 

n=±v0 

or  the  velocity  of  the  wheel  should  be  one-half  that  of  the 
entering  water.  Inserting  this  value,  the  hydraulic  effi- 
ciency corresponding  to  the  advantageous  velocity  is 


t,  —  i  ~~  j 

2gh 


ART.  155  OVERSHOT   WHEELS  417 

and  lastly,  replacing  ^02  by  its  value  2ghQ,  it  becomes 

'-'-IT-!        (155) 

which  is  the  maximum  efficiency  of  the  overshot  wheel. 

This  investigation  shows  that  one-half  of  the  entrance 
fall  h0  and  the  whole  of  the  exit  fall  hl  are  lost,  and  it  is 
hence  plain  that  in  order  to  make  e  as  large  as  possible 
both  hQ  and  hl  should  be  as  small  as  possible.  The  fall  hQ 
is  made  small  by  making  the  radius  of  the  wheel  large; 
but  it  cannot  be  made  zero,  for  then  no  water  would  enter 
the  wheel:  it  is  generally  taken  so  as  to  make  the  angle  6Q 
about  10  or  15  degrees.  The  fall  h^  is  made  small  by  giving 
to  the  buckets  a  form  which  will  retain  the  water  as  long 
as  possible.  As  the  water  really  leaves  the  wheel  at  several 
points  along  the  lower  circumference,  the  value  of  h^  can- 
not usually  be  determined  with  exactness. 

The  practical  advantageous  velocity  of  the  overshot 
wheel,  as  determined  by  the  method  of  Art.  140,  is  found 
to  be  about  o.4V0,  and  its  efficiency  is  found  to  be  high, 
ranging  from  70  to  90  percent.  In  times  of  drought,  when 
the  water  supply  is  low,  and  it  is  desirable  to  utilize  all 
the  power  available,  its  efficiency  is  the  highest,  since  then 
the  buckets  are  but  partly  filled  and  h^  becomes  small. 
Herein  lies  the  great  advantage  of  the  overshot  wheel; 
its  disadvantage  is  in  its  large  size  and  the  expense  of  con- 
struction and  maintenance. 

The  number  of  buckets  and  their  depth  are  governed 
by  no  laws  except  those  of  experience.  Usually  the  num- 
bers of  buckets  is  about  $r  or  6r,  if  r  is  the  radius  of  the 
wheel  in  feet,  and  their  radial  depth  is  from  10  to  15  inches. 
The  breadth  of  the  wheel  parallel  to  its  axis  depends  upon 
the  quantity  of  water  supplied,  and  should  be  so  great  that 
the  buckets  are  not  fully  filled  with  water,  in  order  that 
they  may  retain  it  as  long  as  possible  and  thus  make  hl 


418 


WATER  WHEELS 


CHAP.  XIII 


small.  The  wheel  should  be  set  with  its.  outer  circum- 
ference at  the  level  of  the  tail  water. 

Prob.  155.  Estimate  the  horse-power  and  efficiency  of  an  over- 
shot wheel  which  uses  1080  cubic  feet  of  water  per  minute  under 
a  head  of  26  feet,  the  diameter  of  the  wheel  being  23  feet,  and 
the  water  entering  15°  from  the  top  and  leaving  12°  from  the 
bottom. 

ART.  156.     BREAST  WHEELS 

The  breast  wheel  is  applicable  to  small  falls,  and  the 
action  of  the  water  is  partly  by  impulse  and  partly  by 
weight.  As  represented  in  Fig.  156,  water  from  a 

reservoir  is  admitted 
through  an  orifice  upon 
the  wheel  under  the 
head  h0  with  the  ve- 
locity VQ\  the  water 
being  then  confined 
between  the  vanes  and 
the  curved  breast  acts 
by  its  weight  through 
a  distance  h2t  which  is 
approximately  equal  to 
FlG-156  h-h0,  until  finally  it 

is  released  at  the  level  of  the  tail  race  and  departs  with 
the  velocity  u,  which  is  the  same  as  that  of  the  circum- 
ference of  the  wheel.  The  total  energy  of  the  water  being 
Wh,  the  work  of  the  wheel  is  eWh,  if  e  be  its  efficiency. 

The  reasoning  of  the  last  article  may  be  applied  to  the 
breast  wheel,  hl  being  made  equal  to  zero,  and  the  ex- 
pression there  deduced  for  e  may  be  regarded  as  an  approxi- 
mate value  of  its  theoretic  efficiency.  It  appears,  then, 
that  e  will  be  the  greater  the  smaller  the  fall  h0 ;  but  owing  to 
leakage  between  the  wheel  and  the  curved  breast,  which 
cannot  be  theoretically  estimated,  and  which  is  less  foi 


ART.  153  BREAST  WHEELS  419 

high  velocities  than  for  low  ones,  it  is  not  desirable  to 
make  v0  and  h0  small.  The  efficiency  of  the  breast  wheel 
is  hence  materially  less  than  that  of  the  overshot,  and  usu- 
ally ranges  from  50  to  80  percent,  the  lower  values  being 
for  small  wheels. 

Another  method  of  determining  the  theoretic  efficiency 
of  the  breast  wheel  is  to  discuss  the  action  of  the  water  in 
entering  and  leaving  the  vanes  as  a  case  of  impulse.  Let 
at  the  point  of  entrance  AvQ  and  Au  be  drawn  parallel  and 
equal  to  the  velocities  VQ  and  u,  the  former  being  that  of 
the  entering  water  and  the  latter  that  of  the  vanes.  Let 
a  be  the  angle  between  v0  and  u,  which  may  be  called  the 
angle  of  approach.  Then  the  dynamic  pressure  exerted 
by  the  water  in  entering  upon  and  leaving  the  vanes  is, 
from  Art.  149, 


g 

and  the  work  performed  by  it  per  second  is 


This  expression  has  its  maximum  value  when 

u=%v0  cosa 

which  gives  the  advantageous  velocity  of  the  circumference 
of  the  wheel,  and  the  corresponding  work  of  the  dynamic 
pressure  is 

Vcos^a 

4g 

Adding  this  to  the  work  Wh2  done  by  the  weight  of  the 
water,  the  total  work  of  the  wheel  when  running  at  the 
advantageous  velocity  is  found  to  be 


or,  if  vQ2  be  replaced  by  its  value  c±  .  2ghQ,  where  c^  is  the 


420  WATER  WHEELS  CHAP,  xui 

coefficient  of  velocity  for  the  stream  as  it  leaves  the  orifice 
of  the  reservoir, 


whence  the  maximum  hydraulic  efficiency  of  the  wheel  is 

*=K2cos2a.y°  +  J  (156) 

If  in  this  expression  h2  be  replaced  by  h—hQ,  and  if  cl  =  i 
and  a  =  o°,  this  reduces  to  the  same  value  as  found  for  the 
overshot  wheel.  The  angle  a,  however,  cannot  be  zero> 
for  then  the  direction  of  the  entering  water  would  be  tan- 
gential to  the  wheel,  and  it  could  not  impinge  upon  the 
vanes;  its  value,  however,  should  be  small,  say  from  10° 
to  25°.  The  coefficient  c±  is  to  be  rendered  large  by  making 
the  orifice  of  the  discharge  with  well-rounded  inner  corners. 
so  as  to  avoid  contraction  and  the  losses  incident  thereto. 
The  above  formulas  cannot  be  relied  upon  in  practice  to 
give  close  values  of  k  and  e,  on  account  of  losses  by  foam 
and  leakage  along  the  curved  breast,  which  of  course  can- 
not be  algebraically  expressed. 

Prob.  156.  A  breast  wheel  is  10.5  feet  in  diameter,  and  has 
^  =  0.93,  h0  =  4.2  feet,  and  a  =  12  degrees.  Compute  the  most 
advantageous  number  of  revolutions  per  minute. 

ART.  157.     UNDERSHOT  WHEELS 

The  common  undershot  wheel  has  plane  radial  vanes, 
and  the  water  passes  beneath  it  in  a  direction  nearly  hori- 
zontal. It  may  then  be  regarded  as  a  breast  wheel  where 
the  action  is  entirely  by  impulse,  so  that  in  the  preceding 
equations  hz  becomes  o,  h0  becomes  h,  and  a  will  be  o°. 
The  theoretic  efficiency  then  is  e=%c^.  In  the  best  con- 
tractions the  coefficient  cl  is  nearly  unity,  so  it  may  be 
concluded  that  the  maximum  efficiency  of  the  undershot 
wheel  is  about  0.5.  Experiments  show  tha£  its  actual 
efficiency  varies  from  0.20  to  0.40,  and  that  the  advanta- 


ART.  157  UNDERSHOT    WHEELS.  421 

geous  velocity  is  about  o.4V0  instead  of  o.$v0.  The  lowest 
efficiencies  are  obtained  from  wheels  placed  in  an  unlimited 
flowing  current,  as  upon  a  scow  anchored  in  a  stream;  and 
the  highest  from  those  where  the  stream  beneath  the  wheel 
is  confined  by  walls  so  as  to  prevent  the  water  from  spread- 
ing laterally. 

The  Poncelet  wheel,  so  called  from  its  distinguished 
inventor,  has  curved  vanes,  which  are  so  arranged  that  the 
water  leaves  them  tangentially,  with  its  absolute  velocity 
less  than  that  of  the  velocity  of  the  wheel.  If  in  Fig.  156 
the  fall  h2  be  very  small,  and  the  vanes  be  curved  more 
than  represented,  it  will  exhibit  the  main  features  of  the 
Poncelet  wheel.  The  water  entering  with  the  absolute 


FIG.  157 

velocity  v0  takes  the  velocity  u  of  the  vane  and  the  velocity 
V  relative  to  the  vane.  Passing  then  under  the  wheel,  its 
dynamic  pressure  performs  work ;  and  on  leaving  the  vane 
its  relative  velocity  V  is  probably  nearly  the  same  as  that 
at  entrance.  Then  if  V  be  drawn  tangent  to  the  vane  at 
the  point  of  exit,  and  u  tangent  to  the  circumference,  their 
resultant  will  be  vv  the  absolute  velocity  of  exit,  which 
will  be  much  less  than  u.  Consequently  the  energy  carried 
away  by  the  departing  water  is  less  than  in  the  usual  forms 
of  breast  and  undershot  wheels,  and  it  is  found  by  experi- 
ment that  the  efficiency  may  be  as  high  as  60  percent. 

In  Fig.  157  is  shown  a  portion  of  a  Poncelet  wheel.  At 
A  the  water  enters  the  wheel  through  a  nozzle -like  opening 
with  the  absolute  velocity  VQ  and  at  B  it  leaves  with  the 
absolute  velocity  vlf  In  the  figure  A  and  B  have  the 


422  WATER  WHEELS.  CHAP,  xin 

same  elevation.  At  A  the  entering  stream  makes  the 
approach  angle  a  with  the  circumference  of  the  wheel  and 
the  same  angle  with  the  vane,  so  that  the  relative  velocity 
V  is  equal  to  the  velocity  of  the  outer  circumference  u. 
If  h  be  the  head  on  A  the  theoretic  work  of  the  water  is 
Wh,  and  the  work  of  the  wheel  is 


and  the  efficiency,  neglecting  friction  and  leakage,  is 

v  2  —  v  2 
2gh 

Now,  let  ci  be  the  coefficient  of  velocity  of  the  entrance 
orifice,  then  •v0=c1\/2gh.  From  the  parallelograms  of  ve- 
locity at  A  and  B,  there  are  found 

u=—  v*  =2U  sino:  =v0  tana: 

2  cosa 

and  for  this  velocity  u  the  efficiency  of  the  wheel  is 

€=c±  (i     tan  OL)  (Io7) 

If  cl  =  i  and  a=o,  the  efficiency  becomes  unity.  In  the 
best  constructions  ^  may  be  made  from  0.95  to  0.98,  but  a 
cannot  be  a  very  small  angle,  since  then  no  water  could 
enter  the  wheel.  If  a  =30°  and  £1=0.95  the  efficiency  is 
0.60,  which  is  probably  a  higher  value  than  usually  attained 
in  practice.  If  the  velocity  be  greater  or  less  than  Jz;0/cosa:, 
the  efficiency  will  be  lowered  on  account  of  shock  and 
foam  at  A. 

Prob.  157a.  Estimate  the  horse-power  that  can  be  obtained 
from  an  undershot  wheel  with  plane  radial  vanes  placed  in  a 
stream  having  3r  mean  velocity  of  5  feet  per  second,  the  width 
of  the  wheel  being  15  feet,  its  diameter  8  feet,  and  the  maxi- 
mum immersion  of  the  vanes  being  1.33  feet.  How  many  rev- 
olutions per  minute  should  this  wheel  make  in  order  to  fur- 
nish the  maximum  power?  Make  sketches  showing  how  you 
would  mount  the  wheel  in  the  stream  and  provide  against 
damage  by  floods. 


ART.  158  VERTICAL    IMPULSE    WHEELS  423 

Prob.  1576.  What  width  of  wheel  is  necessary  for  the  data 
of  the  last  problem  in  order  that  75  horse-powers  may  be 
generated  ? 

Prob.  I57c.  Estimate  the  horse-power  that  can  be  obtained 
from  a  Poncelet  wheel  under  a  head  of  4  feet,  when  the  orifice 
at  A  is  2  feet  wide  and  3  inches  deep,  taking  a  =  30°  and 
^  =  0.90. 

ART.  158.     VERTICAL  IMPULSE  WHEELS 

A  vertical  wheel  like  Fig.  157,  Jsut  having  smaller  vanes 
against  which  the  water  is  delivered  from  a  nozzle,  is  often 
called  an  impulse  wheel,  or  a  ' '  hurdy-gurdy ' '  wheel.  The 
Pelton  wheel,  the  Cascade  wheel,  and  other  forms,  can  be 
purchased  in  several  sizes,  and  are  convenient  on  account 
of  their  portability.  Fig.  158a  shows 
an  outline  sketch  of  such  a  wheel  with 
the  vanes  somewhat  exaggerated  in 
size.  The  simplest  vanes  are  radial 
planes  as  at  A ,  but  these  give  a  low 
efficiency.  Curved  vanes,  as  at  B,  are 
generally  used,  as  these  cause  the  water 
to  turn  backward,  opposite  to  the  direc- 
tion of  the  motion,  and  thus  to  leave 
the  wheel  with  a  low  absolute  velocity 
(Art.  150) .  In  the  plan  of  the  wheel  it  is  seen  that  the  vanes 
may  be  arranged  so  as  also  to  turn  the  water  sidewise  while 
deflecting  it  backward.  The  experiments  of  Browne  *  show 
that  with  plane  radial  vanes  the  highest  efficiency  was  40.2 
percent,  while  with  curved  vanes  or  cups  82.5  percent  was 
attained.  The  velocity  of  the  vanes  which  gave  the  highest 
efficiency  was  in  each  case  almost  exactly  one-half  the 
velocity  of  the  jet. 

The  Pelton  wheel  is  used  under  high  heads,  and  also 
being  of  small  size  it  has  a  high  velocity.  The  effective  head 
is  that  measured  at  the  entrance  of  the  nozzle  by  a  pressure 

*  Bowie's  Treatise  on  Hydraulic  Mining  (New  York,  1885),  p.  193. 


424 


WATER  WHEELS 


CHAP.  XIII 


FIG.  1586 


gage,  corrected  for  velocity  of  approach  and  the  loss  in  the 

nozzle  by  formula  (80) j.  These  wheels  are  wholly  of  iron, 

and  are  provided  with 
a  casing  to  prevent  the 
spattering  of  the  water. 
Fig.  1586  shows  a  form 
with  three  nozzles,  by 
which  three  streams 
are  applied  at  different 
parts  of  the  circum- 
ference, in  order  to  ob- 
tain a  greater  power 
than  by  a  single  nozzle, 
or  to  obtain  a  greater 
speed  by  using  smaller 
nozzles.  For  an  effec- 
tive head  of  100  feet 
and  a  single  nozzle 

the  following  quantities  are  given  by  the  manufacturers : 

Diameter  in  feet, 

Cubic  feet  per  minute, 

Revolutions  per  minute, 

Horse-powers, 

and  these  figures  imply  an  efficiency  of  85  percent. 

The  general  theory  of  these  vertical  impulse  wheels  is 
the  same  as  that  given  for  moving  vanes  in  Art.  149.  Owing 
to  the  high  velocity,  more  or  less  shock  occurs  at  entrance, 
and  as  the  angle  of  exit  /?  cannot  be  made  small,  the  water 
leaves  the  vanes  with  more  or  less  absolute  velocity.  The 
advantageous  velocity  of  the  vanes  or  cups  is  between  40 
and  50  percent  of  that  of  the  entering  jet. 

Prob.  158.  The  diameter  of  a  hurdy-gurdy  wheel  is  12.58 
feet  between  centers  of  vanes  and  the  impinging  jet  has  a  ve- 
locity of  58.5  feet  per  second  and  a  diameter  of  0.182  feet.  The 
efficiency  of  the  wheel  is  44.5  percent  when  making  62  revolu- 
tions per  minute.  What  horse-power  does  it  furnish? 


I 

2 

3 

4 

6 

8.29 

44.19 

99-52 

176.7 

398.1 

726 

363 

242 

181 

121 

1  .40 

7-49 

16.84 

29-93 

67.36 

ART.  159 


HORIZONTAL  IMPULSE 


425 


ART.  159.     HORIZONTAL  IMPULSE  WHEELS 

When  a  wheel  is  placed  with  its  plane  horizontal  and  is 
driven  by  a  stream  of  water  from  a  nozzle  it  is  called  a 
horizontal  impulse  wheel.  There  are  twdyformsV  known  as 
the  outward-flow  and  the  inward-flow  wheel.  In  the 
former,  shown  in  Fig.  159a,  the  water  enters  the  wheel  upon 
the  inner  and  leaves  it  upon  the  outer  circumference;  in 
the  latter,  shown  in  Fig.  1596,  the  water  enters  upon  the 
outer  and  leaves  upon  the  inner  circumference.  .The  water 
issuing  from  the  nozzle  with  the  velocity  v  impinges  upon 


FIG.  159a 


FIG.  1596 


the  vanes,  and  in  passing  through  the  wheel  alters  both  its 
direction  and  its  absolute  velocity,  thus  transforming  its 
energy  into  useful  work.  The  energy  of  the  entering  water 
is  W .vz/2g  and  that  of  the  departing  water  is  W .vt2/2g. 
Neglecting  frictional  resistances,  the  work  imparted  to  the 
wheel  by  the  water  is 

k=w(—-v-± 


and  dividing  this  by  the  theoretic  energy,  the  efficiency  is 


This  is  the  same  as  the  general  formula  (154)  if  h'  =o,  that 
is,  if  losses  in  foam  and  friction  are  disregarded,  and  if  the 
wheel  is  set  at  the  level  of  the  tail  race.  It  is  now  required 
to  state  the  conditions  which  will  render  these  losses  and 
also  the  velocity  vl  as  small  as  possible.  The  reasoning  will 


426  WATER  WHEELS  CHAP,  xm 

be  general  and  applicable  to  both  outward-  and  inward- 
flow  wheels. 

At  the  point  A  where  the  water  enters  the  wheel  let  the 
parallelogram  of  velocities  be  drawn,  the  absolute  velocity 
of  entrance  being  resolved  into  its  two  components,  the 
velocity  u  of  the  wheel  at  that  point,  and  the  velocity  V 
relative  to  the  vane ;  let  the  approach  angle  between  u  and 
v  be  called  a,  and  the  entrance  angle  between  u  and  V  be 
called  <t>.  At  the  point  B  where  the  water  leaves  the  wheel 
let  V1  be  its  velocity  relative  to  the  vane,  and  u^  the  veloc- 
ity of  the  wheel  at  that  point ;  then  their  resultant  is  vlt  the 
absolute  velocity  of  exit.  Let  the  exit  angle  between  Vl 
and  the  reverse  direction  of  %  be  called  /?.  The  directions 
of  the  velocities  u  and  ul  are  of  course  tangential  to  the  cir- 
cumferences at  the  points  A  and  B.  Let  r  and  rA  be  the 
radii  of  these  circumferences;  then  the  velocities  of  revo- 
lution are  directly  as  the  radii,  or  ur1  =  ujr. 

In  order  that  the  water  may  enter  the  wheel  without 
shock  and  foam,  the  relative  velocity  V  should  be  tangent 
to  the  vane  at  A,  so  that  the  water  may  smoothly  glide 
along  them.  This  will  be  the  case  if  the  wheel  is  run  at 
such  speed  that  the  parallelogram  at  A  can  be  formed,  or 
when  the  velocities  u  and  v  are  proportional  to  the  sines 
of  the  angles  opposite  them  in  the  triangle  Auv.  The 
velocity  vl  will  be  rendered  very  small  by  running  the  im- 
pulse wheel  at  such  speed  that  the  velocities  u±  and  Vl  are 
equal,  since  then  the  parallelogram  at  B  becomes  a  rhombus, 
and  the  diagonal  v^  is  very  small.  Hence 
u  sm((j>  —  a) 

«-— si*-     and    M'=F-      <159)' 

are  the  two  conditions  of  maximum  hydraulic  efficiency. 

Now,  referring  to  the  formula  (151),  which  expresses 
the  relation  between  the  velocities  of  rotation  and  the 
relative  velocities  of  the  water  for  revolving  vanes,  it  is 
seen  that  if  ul  =  Vlt  then  also  u  =  V.  But  u  cannot  equal 


ART.  159  HORIZONTAL    IMPULSE    WHEELS  427 

V  unless  (j>  =  2a,  and  then  u=v/2  cosa,  which  is  the  advan- 
tageous velocity  of  the  circumference  at  A.  Therefore  the 
two  conditions  above  reduce  to 

<f>  =  2a         and         u= (159), 

2  cosa 

which  show  how  the  wheel  should  be  built  and  what  speed  it 
should  have  to  secure  the  greatest  efficiency.  When  this 
speed  obtains  the  absolute  velocity  v^  is 

fj  i      —  r\  4J        C 11 
U i   —  2.  iAr-\     oJLJ 

and  the  corresponding  hydraulic  efficiency  is 

e  =  i-(-^^-)  (159), 

\r    cosa/ 

by  the  discussion  of  which  proper  values  of  the  approach 
angle  a  and  the  exit  angle  /?  can  be  derived. 

This  formula  shows  that  both  the  approach  angle  a  and 
the  exit  angle  /?  should  be  small  in  order  to  give  high  effi- 
ciency, but  they  cannot  be  zero,  as  then  no  water  could 
pass  through  the  wheel;  values  of  from  15  to  30  degrees  are 
usual  in  practice.  It  also  shows  that  /?  is  more  important 
than  a,  and  if  /?  be  small  a  may  sometimes  be  made  40  or 
45  degrees.  It  likewise  shows  that  for  given  values  of  a 
and  /?  the  inward-flow  wheel,  in  which  rx  is  less  than  r,  has 
a  higher  efficiency  than  the  outward-flow  wheel. 

The  condition  Vl  =  %  renders  the  absolute  exit  velocity 
v^  very  small,  but  it  does  not  give  its  true  minimum.  This 
will  be  obtained  by  making  Vl  =  ul  cos/9,  so  that  the  direction 
of  vl  is  normal  to  that  of  V\,  and  thus  v1=ul  sin/9.  The 
discussion  of  water  wheels  and  turbines  under  this  condi- 
tion of  the  true  minimum  leads  to  very  complex  formulas, 
and  hence  in  this  work,  as  in  many  others,  the  simpler  con- 
dition Vl=ul  is  used. 

Prob.  159a.  Compute  the  maximum  efficiency  of  an  outward- 
flow  impulse  wheel  when  ^  =  3  feet,  r  =  2  feet,  a  =  45°,  ^  =  90°, 
/5  =  30°,  and  find  the  number  of  revolutions  per  minute  required 


428 


WATER  WHEELS 


CHAP.  XIII 


to  secure  such  efficiency  when  the  velocity  of  the  entering  stream 
is  ^  =  100  feet  per  second. 

Prob.  1596.  For  an  inward-flow  impulse  wheel  let  the  angle 
a  =  36°,  the  inner  radius  12  inches,  and  the  outer  radius  21 
inches.  If  the  hydraulic  efficiency  is  0.89,  what  should  be  the 
values  of  the  angle  /?  and  </>?  If  the  velocity  of  the  entering 
jet  is  92  feet  per  second,  what  should  be  the  number  of  revolu- 
tions of  the  wheel  per  minute? 

ART.  160.     DOWNWARD-FLOW  IMPULSE  WHEELS 

In  the  impulse  wheels  thus  far  considered  the  water 
leaves  the  vanes  in  a  horizontal  direction.  Another  form 
used  less  frequently  is  that  of  a  horizontal  wheel  driven 
by  water  issuing  from  an  inclined  nozzle  so  that  it  passes 
downward  along  the  vanes  without  approaching  or  reced- 
ing from  the  axis.  Fig.  160  shows  an  outline  plan  of  such 

an  impulse  wheel  and  a 
'development  of  a  part  of 
a  cylindrical  section.  Let 
v  be  the  velocity  of  the 
entering  stream,  u  that  of 
the  wheel  at  the  point 
where  it  strikes  the  vanes, 
and  vl  the  absolute  veloc- 
ity of  the  departing  water. 
At  the  entrance  A  the 
direction  of  v  makes  with 
that  of  u  the  approach 
angle  a,  and  the  direction 
of  the  relative  velocity  V 
makes  with  that  of  u  the 
entrance  angle  <£.  The 
water  then  passes  over 
the  vane,  and,  neglecting 
FlG- 16°  the  influence  of  friction 

and  gravity,  it  issues  at  B  with  the  same  relative  velocity  V, 
making  the  exit  angle  @  with  the  plane  of  motion. 


ART.  160         DOWNWARD-FLOW  IMPULSE  WHEELS.  429 

The  condition  that  impact  and  foam  shall  be  avoided  at 
A  is  fulfilled  by  making  the  relative  velocity  V  tangent  to 
the  vane,  and  the  condition  that  the  absolute  velocity  vl 
shall  be  small  is  fulfilled  by  making  the  velocities  u  and  V 
equal  at  B.  Hence,  as  in  the  last  article,  the  best  construc- 
tion is  to  make  <£  =20:,  and  the  best  speed  of  the  wheel  is 
u=v/2  cosa.  Also  by  the  same  reasoning  the  efficiency 
-under  these  conditions  is 

e  =  i  —  (sin  J/?/cosa)  2 

which  shows  that  a,  and  especially  /?,  should  be  a  small 
angle  to  give  a  high  numerical  value  of  e.  For  instance,  if 
both  these  angles  are  30  degrees,  the  efficiency  is  0.92,  but 
if  a  =45°  and  /?  =  10°,  the  efficiency  is  0.94. 

Although  these  wheels  are  but  little  used,  there  seems 
to  be  no  hydraulic  reason  why  they  should  not  be  employed 
with  a  success  equal  to  or  greater  than  that  attained  by  ver- 
tical impulse  wheels.  It  will  be  possible  to  arrange  several 
nozzles  around  the  circumference,  and  thus  to  secure  a  high 
power  with  a  small  wheel.  The  fall  of  the  water  through 
the  vertical  distance  between  A  and  B  will  also  add  slightly 
to  the  power  of  the  wheel,  and  if  this  be  taken  into  account 
the  above  values  of  advantageous  velocity  and  efficiency 
will  be  modified,  both  being  slightly  increased,  as  the  follow- 
ing investigation  shows. 

Let  hi  be  the  vertical  fall  between  A  and  B,  then  the 
theoretic  energy  of  the  water  with  respect  to  B  is 

7;2        q  i  2 
hl+—-V-±- 


and  the  hydraulic  efficiency  of  the  wheel  is 

v 


, 

V2  +  2 


Here  the  relative  velocity  V^  at  B  is  greater  than  V,  or 


430  WATER  WHEELS  CHAP,  xin 

and  since  u  should  equal  Vlf  this  equation  becomes,  after 
inserting  for  V  its  value  in  terms  of  u,  v,  and  a, 


H 


v       I        2gh\ 


o     i 

2  cosa  \         v2  ] 

which  gives  the  advantageous  velocity  of  the  wheel.  Since 
v1  =  2U  sinj/?,  the  above  expression  for  the  theoretic  hydraulic 
efficiency  reduces  to 


For  this  case  the  approach  angle  <£  must  be  a  little  greater 
than  2  a,  and  its  value  can  be  found  by 


cot  6  =cota  —   2   . 

v  sin  2a 

and  by  using  this  angle  <£,  losses  due  to  impact  will  be 
avoided  when  the  wheel  is  run  at  the  advantageous  speed. 
For  example,  if  v  =  50  feet  per  second,-  and  h±  =  i  foot,  and 
a  =30°,  the  value  of  <j>  is  about  63°  instead  of  60°  as  the 
simpler  condition  requires,  while  the  increase  in  the  advan- 
tageous speed  is  about  2  percent  over  the  former  value. 

Prob.  L60a.  A  wheel  like  Fig.  160  is  driven  by  water  which 
issues  from  a  nozzle  with  a  velocity  of  100  feet  per  second.  If 
the  diameter  is  3  feet,  the  efficiency  0.90,  and  the  approach 
angle  01=45  degrees,  find  the  best  values  of  the  entrance  and 
exit  angles  and  the  best  speed. 

Prob.  1606.  For  a  wheel  of  the  same  dimensions  and  data. 
let  the  vertical  fall  h^  be  1.25  feet.  Compute  the  entrance  and 
exit  angles  and  the  best  speed.  If  the  discharge  from  the 
nozzle  is  0.87  cubic  feet  per  second,  what  is  the  horse-power  of 
the  wheel. 

ART.  161.     NOZZLES  FOR  IMPULSE  WHEELS 

s 

Impulse  wheels  are  driven  by  the  dynamic  pressure  of 
water  issuing  from  nozzles  attached  to  the  end  of  a  pipe 
which  conducts  the  water  from  a  reservoir.  It  is  shown 


ART.  161  NOZZLES  FOR  IMPULSE  WHEELS  431 

in  Art.  97  that  the  greatest  velocity  is  secured  when  the 
diameter  of  the  nozzle  is  as  small  as  possible  and  that  the 
greatest  discharge  occurs  when  there  is  no  nozzle.  To 
secure  the  greatest  power,  however,  there  is  a  certain 
diameter  of  nozzle  which  will  now  be  determined,  and  it  is 
advisable  for  economical  reasons  to  use  a  nozzle  of  this  size 
and  adjust  the  speed  of  the  wheel  thereto. 

Let  h  be  the  hydrostatic  head  on  the  nozzle,  I  the  length, 
and  d  the  diameter  of  the  pipe,  and  D  the  diameter  of  the 
nozzle.  Let  all  the  resistances  except  that  due  to  friction 
in  the  pipe  and  nozzle  be  neglected;  then  from  Art.  97, 
the  velocity  of  the  jet  from  the  nozzle  is 


2gh 


in  which  /  is  the  friction  factor  for  the  pipe  and  cl  is  the 
coefficient  of  velocity  for  the  nozzle.  Let  w  be  the  weight 
of  a  cubic  foot  of  water;  then  the  theoretic  energy  of  the 
jet  per  second  is 


and  the  value  of  D  which  renders  this  a  maximum  is,  by 
the  usual  method  of  differentiation,  ascertained  to  be 


^  • 


and  for  a  nozzle  of  this  size  the  velocity  of  the  jet  is 
V  =o.8i6<;1\/  2gh 

or,  since  c±  is  about  0.97,  the  velocity  of  the  jet  when  leaving 
the  nozzle  is  about  80  percent  of  the  theoretic  velocity  due 
to  the  head  on  the  nozzle. 

As  an  example  let  a  pipe  be  1200  feet  long  and  one  foot 
in  diameter  ;  then,  taking  for  /  the  mean  value  0.02  and 
using  c1  =cu£]j  there  is  found  17=0.39  feet,  and  hence  a 


432  WATER  WHEELS  CHAP.XIII 

nozzle  4!  inches  in  diameter  is  required  to  give  the  maxi- 
mum power.  This  result  may  be  revised,  if  thought  neces- 
sary, by  finding  the  velocity  in  the  pipe  and  thus  getting  a 
better  value  of  /  from  Table  33.  If  the  head  be  100  feet, 
this  velocity  is  found  to  be  9.2  feet  per  second,  whence 
/=o.oi8,  and  on  repeating  the  computation  there  is  found 
1^=0.40  feet  =4.  8  inches.  If  the  pipe  be  12  ooo  feet  long, 
the  advantageous  diameter  of  the  nozzle  will  be  found  to 
be  much  smaller,  namely  i\  inches. 

When  there  is  more  than  one  nozzle  at  the  end  of  the 
pipe  the  above  investigation  must  be  modified.  Let  there 
be  two  nozzles  with  the  diameters  D^  and  D2,  each  having 
the  coefficient  c^.  Then  the  discharge  \Tid2v  through  the 
pipe  equals  the  discharge  ^n(D12V1  +  D22V2).  But  the  ve- 
locities Y!  and  V2  are  equal  if  the  tips  of  the  nozzles  are  on 
the  same  elevation,  and  hence  d2v  equals  (Df  +  D^V, 
where  V  is  the  velocity  of  flow  from  each  nozzle.  Now, 
referring  to  Art.  97  and  to  the  proof  of  (161),  it  is  seen  that 
it  applies  to  this  case  provided  D2  be  replaced  by  D^-\~D22, 
and  accordingly 

c12l)*  (161), 


is  the  formula  for  determining  the  sizes  of  the  two  nozzles 
which  will  furnish  the  maximum  power  ;  if  Dl  be  assumed, 
the  value  of  D2  can  be  computed.  The  area  of  the  circle 
of  diameter  D  found  from  (161^  is  equal  to  the  sum  of  the 
areas  of  the  two  circles  found  from  (161)  2.  If  there  be 
three  or  more  nozzles,  the  sum  of  their  areas  is  equal  to 
that  corresponding  to  the  diameter  D  as  computed  from 
(161)!.  For  example,  let  there  be  a  pipe  1200  feet  long 
and  one  foot  in  diameter  to  which  three  nozzles  of  equal 
size  are  attached.  The  diameter  found  above  for  one  noz- 
zle is  4.80  inches,  and  the  corresponding  area  is  18.10  square 
inches;  hence  the  area  of  the  cross-section  of  the  tip  of 
each  of  the  three  nozzles  is  6.03  square  inches,  which  corre- 
sponds to  a  diameter  of  2.77  inches. 


ART.  162  SPECIAL   FORMS   OF   WHEELS  433 

Prob.  161a.  A  pipe  15  ooo  feet  long  and  18  inches  in  diam- 
eter runs  from  a  mountain  reservoir  to  a  power  plant,  where  the 
water  is  to  be  delivered  through  two  nozzles  against  a  hurdy- 
gurdy  wheel.  If  the  diameter  of  one  nozzle  is  2  inches,  find  the 
diameter  of  the  other  in  order  that  the  maximum  power  may 
be  developed.  If  the  head  on  the  nozzles  is  623  feet  and  the 
efficiency  of  the  wheel  79  percent,  compute  the  horse-power 
that  may  be  expected. 

Prob.  1616.  A  compound  pipe  of  lengths  ^  and  12  and  diam- 
eters d^  and  d2  conveys  water  to  an  impulse  wheel  against  which 
it  is  delivered  by  three  nozzles  having  the  diameters  D^  D2, 
and  D3,  the  tips  of  which  are  on  the  same  elevation.  Taking 
the  coefficients  of  velocity  of  the  nozzles  as  equal  and  regarding 
the  diameters  D^  and  D2  as  known,  find  the  diameter  D3  that 
will  render  the  energy  of  the  jets  a  maximum. 


ART.  162.     SPECIAL  FORMS  OF  WHEELS 

Numerous  varieties  of  the  water  wheels  above  described 
have  been  used,  but  the  variation  lies  in  mechanical  details 
rather  than  in  the  introduction  of  any  new  hydraulic  princi- 
ples. In  order  that  a  wheel  may  be  a  success  it  must  fur- 
nish power  as  cheaply  as  or  cheaper  than  steam  or  other 
motors,  and  to  this  end  compactness,  durability,  and  low 
cost  of  installation  and  maintenance  are  essential. 

A  variety  of  the  overshot  wheel,  called  the  back-pitch 
wheel,  has  been  built  in  which  the  water  is  introduced  on 
the  back  instead  of  on  the  front  of  the  wheel.  The  buckets 
are  hence  differently  arranged  from  those  of  the  usual  form, 
and  the  wheel  revolves  also  in  an  opposite  direction.  One 
of  the  largest  overshot  wheels  ever  constructed  is  at  Laxey, 
on  the  west  coast  of  England.  It  is  72  J  feet  in  diameter, 
about  10  feet  in  width,  and  furnishes  about  150  horse-power, 
which  is  used  for  pumping  water  out  of  a  mine. 

A  breast  wheel  with  very  long  curved  vanes  extending 
over  nearly  a  fourth  of  the  circumference  has  been  used  for 


434  WATER  WHEELS  CHAP,  xin 

small  falls,  the  water  entering  directly  from  the  penstock 
without  impulse,  so  that  the  action  is  that  of  weight  alone. 
This  form  is  made  of  iron  and  gives  a  high  efficiency. 

Undershot  wheels  with  curved  floats  for  use  in  the  open 
current  of  a  river  have  been  employed,  but  in  order  to  obtain 
much  power  they  require  to  be  large  in  size,  and  hence  have 
not  been  able  to  compete  with  other  forms.  The  great 
amount  of  power  wasted  in  all  rivers  should,  however,  incite 
inventors  to  devise  wheels  that  can  economically  utilize  it. 
Currents  due  to  the  movement  of  the  tides  also  afford  oppor- 
tunity for  the  exercise  of  inventive  talent. 

The  conical  wheel,  or  Danaide,  is  an  ancient  form  of 
downward-flow  impulse  wheel,  in  which  the  water  approaches 
the  axis  as  it  descends,  and  thus  its  relative  motion  is  de- 
creased by  the  centrifugal  force.  The  theory  of  this  is 
almost  precisely  the  same  as  that  of  an  inward-flow  impulse 
wheel,  and  there  seems  to  be  no  hydraulic  reason  why  it 
should  not  give  a  high  efficiency.  Another  form  of  danaide 
has  two  or  more  vertical  vanes  attached  to  an  axis,  which 
are  enclosed  in  a  conical  case  to  prevent  the  lateral  escape 
of  the  water. 

A  water-pressure  engine  is  a  hydraulic  motor  which 
moves  under  the  static  pressure  of  water  acting  against  a 
piston  or  a  revolving  disk.  The  piston  forms  are  recipro- 
cating in  motion  like  the  steam-engine  and  operate  in  the 
same  way,  the  water  entering  and  leaving  through  ports 
which  are  opened  and  closed  by  a  link  motion  connected 
with  the  piston-rod.  The  other  forms  give  rotary  motion 
directly  from  the  revolving  vanes  or  disks.  The  piston 
engine  has  been  employed  in  Germany  to  a  considerable 
extent  to  drive  pumps  for  draining  mines,  but  the  rotary 
engine  has  not  been  widely  used  and  it  cannot  be  advan- 
tageously arranged  to  deliver  a  high  power.  On  account  of 
the  incompressibility  of  water  special  devices  for  regulating 
the  opening  and  closing  of  the  valves  are  necessary. 


ART.  162  SPECIAL   FORMS    OF   WHEELS  435 

Numerous  other  special  devices  for  utilizing  the  energy 
of  water  by  means  of  water  wheels  have  been  invented,  but 
they  do  not  introduce  any  new  hydraulic  principle.  The 
efficiency  of  these  special  forms  is  often  low  on  account  of 
the  imperfections  of  the  apparatus,  but  it  should  be  borne 
in  mind  that  high  efficiency  is  only  obtained  after  trials 
extending  over  much  time,  such  trials  enabling  the  imper- 
fections to  be  discovered  and  removed.  The  formulas  for 
hydraulic  efficiency  deduced  in  the  preceding  pages  do  not  in- 
clude losses  due  to  friction,  and  these  may  often  amount  to 
ten  or  twenty  percent  of  the  theoretic  energy,  so  that  due 
allowance  for  them  should  be  made  in  estimating  the  power 
which  a  proposed  design  may  deliver. 

Power  may  be  obtained  from  the  ocean  waves,  which  are 
constantly  rising  and  falling,  by  a  suitable  arrangement  of 
wheels  and  levers,  and  some  inventions  in  this  direction  have 
given  fair  promise  of  success.  One  in  operation  on  the 
coast  of  England  about  1890  consisted  of  a  large  buoy 
which  rose  and  fell  with  the  waves  on  a  fixed  vertical  shaft 
fastened  in  the  rock  bottom.  As  the  buoy  moved  up  and 
down  it  operated  a  system  of  levers  and  wheels  which  drove 
an  air-compressor,  and  this  in  turn  ran  a  dynamo  that  gen- 
erated electric  power.  The  rise  of  the  ocean  tide  also  affords 
opportunity  for  impounding  water  which  may  be  used  to 
generate  power  when  the  tide  falls.  Plants  for  this  pur- 
pose are  to  be  located  along  tidal  rivers  where  opportuni- 
ties for  impounding  occur,  the  wheels  being  idle  during  the 
rise  of  the  tide  and  in  operation  during  its  fall.  Owing  to 
this  intermittent  generation  of  power,  it  will  be  necessary 
to  provide  for  its  storage,  so  that  industries  using  it  may 
be  in  continuous  operation. 

Prob.  I62a.  A  wheel  using  10.5  cubic  meters  of  water  per 
minute  under  an  effective  head  of  23.4  meters  has  an  efficiency 
of  75  percent.  What  metric  horse-power  does  it  deliver?  What 
is  its  power  in  kilowatts? 


436  WATER  WHEELS  CHAP,  xin 

Prob.  1626.  A  breast  wheel  has  ^  =  0.95,  h0=i.^  meters,  and 
a  =  12  degrees.  If  its  diameter  is  3.5  meters,  compute  the  most 
advantageous  number  of  revolutions  per  minute. 

Prob.  162c.  An  inward-flow  impulse  wheel  has  0  =  104°, 
a  =  52°,  and  /?  =  i2°,  its  inner  diameter  being  0.82  meters  and 
its  outer  diameter  1.22  meters.  If  this  wheel  uses  0.86  cubic 
meters  of  water  per  second  under  an  effective  head  of  7.9  meters, 
compute  its  efficiency  and  its  probable  horse-power. 

Prob.  162d.  A  pipe  3200  meters  long  and  40  centimeters  in 
diameter  delivers  water  through  two  nozzles  against  a  hurdy- 
gurdy  wheel.  When  the  diameter  of  one  nozzle  is  5  centimeters 
find  the  diameter  of  the  other  nozzle  in  order  that  the  energy 
of  the  two  jets  may  be  a  maximum.  If  the  head  on  the  nozzles 
is  107  meters  and  the  efficiency  of  the  wheels  is  81  percent,  com- 
pute the  horse-power  which  the  wheels  will  deliver. 


ART.  163 


THE  REACTION  WHEEL 


437 


CHAPTER    XIV 
TURBINES 

ART.  163.     THE  REACTION  WHEEL 

The  reaction  wheel,  invented  by  Barker  about  1740, 
consists  of  a  number  of  hollow  arms  connected  with  a  hollow 
vertical  shaft,  as  shown  in  Fig.  163.  B 
The  water  issues  from  the  ends  of  the 
arms  in  a  direction  opposite  to  that 
of  their  motion,  and  by  the  dynamic 
pressure  due  to  its  reaction  the  energy 
of  the  water  is  transformed  into  useful 
work.  Let  the  head  of  water  CC  in 
the  shaft  be  h ;  then  the  pressure-head 
BE  which  causes  the  flow  from  the 
arms  is  greater  than  h,  on  account  of 
the  centrifugal  force  due  to  the  rota- 
tion of  the  wheel.  Let  %  be  the  abso- 
lute velocity  of  the  exit  orifices,  and 
V1  be  the  velocity  of  discharge  relative 
to  the  wheel;  then,  as  shown  in  Art. 
31,  and  also  in  Art.  153,  FJG.  153 


N 

B 

i 
i 

T 

[B    A               || 

[]>! 

Iti 


The  absolute  velocity  vl  of  the  issuing  water  now  is 
vl  =  Vl-ul=  \/2gh 


It  is  seen  at  once  that  the  efficiency  can  never  reach  unity 
unless  ^!=o,  which  requires  that  Vl=ul.  This,  however, 
can  only  occur  when  ul  =  °o  ,  since  the  above  formula  shows 


438  TURBINES  CHAP,  xiv 

that  F!  must  be  greater  than  n^  for  any  finite  values  of  h 
and  u^.  To  deduce  an  expression  for  the  efficiency  the  work 
of  the  wheel  W(h  —v^/2g)  is  to  be  divided  by  the  theoretic 
energy  of  the  water  Wh,  and  this  gives 


which  shows,  as  before,  that  e  equals  unity  when  Vl=ul=  oo. 
If  Fj  =  2%,  the  value  of  e  is  0.667  ;  if  Vl=^ult  the  value  of  e 
is  reduced  to  0.50. 

This  investigation  indicates  that  the  efficiency  of  a  reac- 
tion wheel  increases  with  its  speed.  If  at  be  the  area  of  the 
exit  orifices  and  w  the  weight  of  a  cubic  unit  of  water,  the 
weight  of  the  water  discharged  in  one  second  is  wa^V^  which 
becomes  infinite  when  Vl=u1=  oo.  Nothing  approaching 
this  can  be  realized,  and  on  account  of  losses  due  to  friction, 
a1  very  high  speed  is  impracticable.  The  reaction  wheel, 
indeed,  is  like  the  jet  propeller  (Art.  177). 

To  consider  the  effect  of  friction  in  the  arms,  let  cl  be  the 
coefficient  of  velocity  (Chapter  VII),  so  that 


Then  the  effective  work  of  the  wheel  is 


s 

and  the  corresponding  efficiency  of  the  wheel  is 


The  value  of  ult  which  renders  this  a  maximum,  is 


and  this  reduces  the  value  of  the  efficiency  to 


ART.  164  CLASSIFICATION    OF   TURBINES  439 

If  G!  =  i,  there  is  no  loss  in  friction,  and  u^  =  oo  and  e  =  i,  as 
before  deduced.  If  c^  =0.94,  the  advantageous  velocity  ul 
is  very  nearly  \/2gh,  and  e  is  0.66 ;  hence  the  influence  of 
friction  in  diminishing  the  efficiency  is  very  great.  In  order 
to  make  ct  large,  the  end  of  the  arm  where  the  water  enters 
must  be  well  rounded  to  prevent  contraction,  and  the  in- 
terior surface  must  be  smooth.  If  the  inner  end  has  sharp 
square  edges,  as  in  a  standard  tube  (Art.  76),  c1  is  0.82,  and 
e  becomes  0.43. 

The  reaction  wheel  is  not  now  used  as  a  hydraulic  motor 
on  account  of  its  low  efficiency.  Even  when  run  at  high  speeds 
the  efficiency  is  low  on  account  of  the  greater  friction  and 
resistance  of  the  air.  By  experiments  on  a  wheel  one  meter 
in  diameter  under  a  head  of  1.3  feet  Weisbach  found  a  maxi- 
mum efficiency  of  67  percent  when  the  velocity  of  revolution 
u±  was  \/2gh.  When  u^  was  2\/2gh  the  efficiency  was  noth- 
ing, or  all  the  energy  was  consumed  in  frictional  resistances. 

The  reaction  wheel  is  here  introduced  at  the  beginning 
of  the  discussion  of  turbines  mainly  to  call  attention  to  the 
fact  that  the  discharge  varies  with  the  speed.  Although 
sometimes  called  a  turbine,  it  can  scarcely  be  properly  con- 
sidered as  belonging  to  that  class  of  motors. 

Prob.  163.  The  sum  of  the  exit  orifices  of  a  reaction  wheel 
is  4.25  square  inches,  their  radius  is  1.75  feet,  and  their  velocity 
32.1  feet  per  second.  Compute  the  head  necessary  to  furnish 
.1.6  horse-powers,  when  ^  =  0.95. 

ART.  164.     CLASSIFICATION  OF  TURBINES 

A  turbine  wheel  may  be  defined  as  one  in  which  the 
water  enters  around  the  entire  circumference  instead  of 
upon  one  portion,  so  that  all  the  moving  vanes  are  simul- 
taneously acted  upon  by  the  dynamic  pressure  of  the  water 
as  it  changes  its  direction  and  velocity.  The  turbine  was 
invented  by  Fourneyron  in  1827,  and  owing  to  its  compact- 


440  TURBINES  CHAP,  xiv 

ness,  cheapness,  and  high  efficiency  it  has  largely  replaced 
the  older  forms  of  water  wheels.  Turbines  are  usually  hori- 
zontal wheels,  and  like  the  impulse  wheels  of  the  last  chap- 
ter, they  may  be  outward-flow,  inward-flow,  or  downward- 
flow,  with  respect  to  the  manner  in  which  the  water  passes 
through  them.  In  the  outward-flow  type  the  water  enters 
the  wheel  around  the  entire  inner  circumference  and  passes 
out  around  the  entire  outer  circumference  (Fig.  1656).  In 
the  inward-flow  type  the  motion  is  the  reverse  (Fig.  165c). 
In  the  downward-flow  type  the  water  enters  around  the 
entire  upper  annular  openings,  passes  downward  between 
the  moving  vanes,  and  leaves  through  the  lower  annulus 
(Fig.  170a).  In  all  cases  the  water  in  leaving  the  wheel 
should  have  a  low  absolute  velocity,  so  that  most  of  its 
energy  may  be  surrendered  to  the  turbine  in  the  form  of 
useful  work. 

The  supply  of  water  to  a  turbine  is  regulated  by  a  gate 
or  gates,  which  can  partially  or  entirely  close  the  orifices 
where  the  water  enters  or  leaves.  The  guides  and  wheel, 
with  the  gates  and  the  surrounding  casings,  are  made  of  iron. 
Numerous  forms  with  different  kinds  of  gates  and  different 
proportions  of  guides  and  vanes  are  in  the  market.  They 
are  made  of  all  sizes  from  6  to  60  inches  in  diameter,  and 
larger  sizes  are  built  for  special  cases.  The  great  turbines 
at  Niagara  are  of  the  outward-flow  type,  the  inner  diameter 
of  a  wheel  being  63  inches  and  each  twin  turbine  furnishing 
about  5000  horse-powers.  The  smaller  sizes  of  turbines 
used  in  the  United  States  are  mostly  of  the  inward-flow  type 
or  of  a  combined  inward-  and  downward-flow  type. 

The  three  typical  classes  of  turbines  above  described  are 
often  called  by  the  names  of  those  who  first  invented  or  per- 
fected them ;  thus  the  outward-flow  is  called  the  Fourneyron, 
the  inward-flow  the  Francis,  and  the  downward-flow  the 
Jonval  turbine.  There  are  also  many  turbines  in  the  market 
in  which  the  flow  is  a  combination  of  inward  and  downward 


ART.  164  CLASSIFICATION    OF   TURBINES  441 

motion,  the  water  entering  horizontally  and  inward,  and 
leaving  vertically,  the  vanes  being  warped  surfaces.  The 
usual  efficiency  of  turbines  at  full  gate  is  from  70  to  85  per- 
cent, although  90  percent  has  in  some  cases  been  derived. 
When  the  gate  is  partly  closed  the  efficiency  in  general  de- 
creases, and  when  the  gate  opening  is  small  it  becomes  very 
low.  This  is  due  to  the  loss  of  head  consequent  upon  the 
sudden  change  of  cross-section;  and  therein  lies  the  disad- 
vantage of  the  turbine,  for  when  the  water  supply  is  low,  it 
is  important  that  it  should  utilize  all  the  power  available. 

Another  classification  is  into  impulse  and  reaction  tur- 
bines. In  an  impulse  turbine  the  water  enters  the  wheel 
with  a  velocity  due  to  the  head  at  the  point  of  entrance, 
just  as  it  does  from  the  nozzle  which  drives  an  impulse  wheel 
(Art.  159).  In  a  reaction  turbine,  however,  the  velocity  of 
the  entering  water  may  be  greater  or  less  than  that  due  to 
the  head  on  the  orifices  of  entrance,  and,  as  in  the  reaction 
wheel,  it  is  also  influenced  by  the  speed.  This  is  due  to  the 
fact  that  in  a  reaction  turbine  the  static  pressure  of  the  water 
is  partially  transmitted  into  the  moving  wheel,  provided 
that  the  spaces  between  the  vanes  are  fully  filled.  Any 
turbine  may  be  made  to  act  either  as  an  impulse  or  a  reac- 
tion turbine.  If  it  be  arranged  so  that  the  water  passes 
through  the  vanes  without  filling  them,  it  is  an  impulse  tur- 
bine ;  if  it  be  placed  under  water,  or  if  by  other  means  the 
flowing  water  is  compelled  to  completely  fill  all  the  passages, 
it  acts  as  a  reaction  turbine.  As  will  be  seen  later,  the 
theory  of  the  reaction  turbine  is  quite  different  from  that 
of  the  impulse  turbine. 

Prob.  164a.  If  the  efficiency  of  a  turbine  is  75  percent  when 
delivering  5000  horse-powers  under  a  head  of  136  feet,  how 
many  cubic  feet  of  water  per  minute  pass  through  it? 

Prob.  1646.  An  outward-flow  turbine  has  a  diameter  of 
3.317  feet.  What  is  the  velocity  of  the  circumference  when  the 
number  of  revolutions  per  minute  is  86  ? 


442 


TURBINES 


CHAP.  XIV 


ART.  165.     REACTION  TURBINES 

A  reaction  turbine  is  driven  by  the  dynamic  pressure  of 
flowing  water  which  at  the  same  time  may  be  under  a  cer- 
tain degree  of  static  pressure.  If  in  the  reaction  wheel  of 
Fig.  163  the  arms  be  separated  from  the  penstock  at  A ,  and 
be  so  arranged  that  BA  revolves  around  the  axis  while  AC 
is  stationary,  the  resulting  apparatus  may  be  called  a  reac- 
tion turbine.  The  static  pressure  of  the  head  CC  can  still 
be  transmitted  through  the  arms,  so  that,  as  in  the  reaction 
wheel,  the  discharge  will  be  influenced  by  the  speed  of  rota- 
tion. The  general  arrangement  of  the  moving  part  is,  how- 
ever, like  that  of  an  impulse  wheel,  the  vanes  being  set 
between  two  annular  frames,  which  are  attached  by  arms 
to  a  central  axis,  In  Fig.  165a  is  a  vertical  section  showing 


FIG.  165a 


an  outward-flow  wheel  W  to  which  the  water  is  brought  by 
guides  G  from  a  fixed  penstock  P.  Between  the  guides  and 
the  wheel  there  is  an  annular  space  in  which  slides  an  an- 
nular vertical  gate  E ;  this  serves  to  regulate  the  quantity 
of  water,  and  when  it  is  entirely  depressed  the  wheel  stops. 


ART.  165 


REACTION  TURBINES 


443 


Many  other  forms  of  gates  are,  however,  used  in  the  different 
styles  of  turbines  found  in  the  market. 


FIG.  165<;. 


FIG.  1656. 


In  the  following  figures  are  given  horizontal  and  vertical 
sections  of  both  the  outward-  and  the  in  ward-flow  types,  show- 


FIG.  lQ5d 

ing  the  arrangement  of  guides  and  vanes.     The  fixed  guide 
passages  which  lead  the  water  from  the  penstock  are  marked 


444 


TURBINES 


CHAP.  XIV 


G,  while  the  moving  wheel  is  marked  W.  It  is  seen  that 
the  water  is  introduced  around  the  entire  circumference  of 
the  wheel,  and  hence  the  quantity  supplied,  and  likewise  the 
power,  is  far  greater  than  in  the  impulse  wheels  of  the  last 
chapter. 

In  order  that  the  static  pressure  may  be  transmitted 
into  the  wheel  it  is  placed  under  water,  as  in  Fig.  165a,  or 
the  exit  orifices  are  partially  closed  by  gates,  or  the  air  is 
prevented  from  entering  them  by  some  other  device. 

In  Fig.  165d  a  Leffel  turbine  of  the  inward-flow  type  is 
illustrated,  the  arrows  showing  the  direction  of  the  water 

as  it  enters  and  leaves.  The 
wheel  itself  is  not  visible,  it 
being  within  the  enclosing 
case  through  which  the  water 
enters  by  the  spaces  between 
the  guides.  In  Fig.  1650  is 
shown  a  view  of  a  Hunt  tur- 
bine, which  is  also  of  the 
inward-  and  downward-flow 
type.  In  both  cases  the  guides 
are  seen  with  the  small  shaft 
for  moving  the  gates,  these 
being  partly  raised  in  Fig. 
165e.  The  flange  at  the  base 
of  the  guides  serves  to  support 
the  weight  of  the  entire  ap- 
paratus upon  the  floor  of  the 
enclosing  penstock,  which  is 
FIG.  1650  filled  with  water  to  the  level 

of  the  head  bay.  The  cylinder  below  the  flange,  commonly 
called  a  draft-tube,  carries  away  the  water  from  the  wheel, 
and  the  level  of  the  tail  water  should  stand  a  little  higher 
than  its  lower  rim  in  order  to  prevent  the  introduction  of 
air,  and  thus  ensure  that  the  wheel  may  act  as  a  reaction 


ART.  165 


REACTION  TURBINES 


445 


turbine.  Iron  penstocks  are  frequently  used  instead  of 
wooden  ones,  and  for  the  pure  outward-  and  inward-flow 
types  the  wheel  is  often  placed  below  the  level  of  the  tail 
race. 

Turbines  are  sometimes  placed  vertically  on  a  horizontal 
shaft.  Fig.  165/  shows  twin  Eureka  turbines  thus  arranged 
in  an  enclosing  iron  casing.  The  water  enters  through  a 


FIG.  165/ 

large  pipe  attached  to  the  cylinder  opening,  and  having 
filled  the  cylindrical  casing  it  passes  through  the  guides, 
turns  the  wheels,  and  escapes  by  the  two  elbows.  Large 
twin  vertical  turbines  furnishing  1200  horse-powers  have 
been  built  by  the  James  Leffel  Company. 

All  reaction  turbines  will  act  as  impulse  turbines  when 
from  any  cause  the  passages  between  the  vanes,  or  buckets, 
as  they  are  generally  called,  are  not  filled  with  water.  In 
this  case  the  theory  of  their  action  is  exactly  like  that  of  the 
impulse  wheels  described  in  the  last  chapter.  In  Arts.  166- 
169  reaction  turbines  of  the  simple  outward-  and  inward- 
flow  types  will  be  discussed,  the  downward-flow  type  being 
reserved  for  special  description  in  Art.  170. 

Prob.  165a.  Consult  Engineering  Record,  Feb.  5,  1898,  and 
describe  methods  of  regulating  the  speed  of  turbines. 

Prob.  1656.  Consult  Bodmer's  Hydraulic  Motors,  Slagg's 
Water  or  Hydraulic  Motors,  and  Weisbach's  Mechanics  of  Engi- 
neering, vol.  2 ;  make  sketches  showing  several  different  arrange- 
ments of  the  gates  of  turbines. 


446 


TURBINES 


CHAP.  XIV 


ART.  166.     FLOW  THROUGH  REACTION  TURBINES 

The  discharge  through  an  impulse  turbine,  like  that  for 
an  impulse  wheel,  depends  only  on  the  area  of  the  guide  ori- 
fices and  the  effective  head  upon  them,  or  q=av=a\/2gh. 
In  a  reaction  turbine,  however,  the  discharge  is  influenced 
by  the  speed  of  revolution,  as  in  the  reaction  wheel,  and 
also  by  the  areas  of  the  entrance  and  exit  orifices.  To  find 
an  expression  for  this  discharge  let  the 
wheel  be  supposed  to  be  placed  below 
the  surface  of  the  tail  water,  as  in  Fig. 
166.  Let  h  be  the  total  head  between 
the  upper  water  level  and  that  in  the 
tail  race,  H^  the  pressure-head  on  the 
exit  orifices,  and  H  the  pressure-head 
at  the  gate  opening  as  indicated  by  a 
_  piezometer  supposed  to  be  there  in- 

FIG.  166  serted.     Let  ui  and  u  be  the  velocities 

of  the  wheel  at  the  exit  and  entrance  circumference,  which 
have  radii  rl  and  r  (Fig.  1656)  .  Let  Vl  and  V  be  the  relative 
velocities  of  exit  and  entrance,  and  v0  be  the  absolute  ve- 
locity of  the  water  as  it  leaves  the  guides  and  enters  the 
wheel;  the  entering  velocity  v0  may  be  less  or  greater  than 
\/2gh,  depending  upon  the  value  of  the  pressure  -head  H. 
Let  ax,  a,  and  a0  be  the  areas  of  the  orifices  normal  to  the 
directions  of  V\,  V,  and  VQ.  Now,  neglecting  all  losses  of 
friction  between  the  guides,  the  theorem  of  Art.  32,  that 
pressure-head  plus  velocity-head  equals  the  total  head, 
gives  the  equation 


Also,  neglecting  the  friction  and  foam  in  the  buckets,  the 
corresponding  theorem  of  Art.  153  gives 


2g        2g 


2g         2g 


ART.  166  FLOW   THROUGH    REACTION    TURBINES  447 

Adding  these  equations,  the  pressure-heads  H  l  and  H  disap- 
pear, and  there  results  the  formula 

VS  -V2  +  vQ2  =  2gh  +  u^  -u2  (166)! 

Now,  since  the  buckets  are  fully  filled,  the  same  quantity  of 
water,  q,  passes  in  each  second  through  each  of  the  areas  alt 
a,  and  a0,  and  hence  the  three  velocities  through  these  areas 
have  the  respective  values, 


Introducing  these  values  into  the  formula  (166)^  solving  for 
q,  and  multiplying  by  a  coefficient  c  to  account  for  losses  in 
leakage  and  friction,  the  discharge  per  second  is 


1=c   FF  (166), 


This  is  the  formula  for  the  flow  through  a  reaction  turbine 
when  the  gate  is  fully  raised.  The  reasoning  applies  to  an 
inward-flow  as  well  as  to  an  outward-flow  wheel.  In  an 
outward-flow  turbine  u^  is  greater  than  u,  and  consequently 
the  discharge  increases  with  the  speed ;  in  an  inward -flow 
turbine  ^is  less  than  u,  and  consequently  the  discharge  de- 
creases as  the  speed  increases. 

The  value  of  the  coefficient  c  will  probably  vary  with  the 
head,  and  also  with  the  size  of  the  areas  alt  a,  and  a0.  When 
a  turbine  has  been  tested  by  the  methods  of  Arts.  138-141, 
and  the  areas  have  been  measured,  the  values  of  c  for  dif- 
ferent speeds  may  be  computed.  For  example,  take  the 
outward-flow  Boyden  turbine,  tests  of  which  at  full  gate  are 
given  in  Art.  141.  The  measured  dimensions  and  angles  of 
this  wheel  are  as  follows : 

Outer  radius  of  wheel  rl  =  3 . 3 1 6  7  feet 

Inner  radius  of  wheel  r  =2.6630  feet 

Outer  radius  of  guide  case  r0  =  2 . 59 1 1  feet 


448  TURBINES  CHAP,  xiv 

Outer  depth  of  buckets  ^  =  0.722  feet 

Inner  depth  of  buckets  d  =0.741  feet 

Outer  area  of  buckets  ax  =  4.61  square  feet 

Inner  area  of  buckets  a  =12.12  square  feet 

Outer  area  of  guide  orifices  0^  =  4.76  square  feet 

Exit  angle  of  buckets  /?  =13.5  degrees 

Entrance  angle  of  buckets  (j>  =  90  degrees 

Entrance  angle  of  guides  a  =24  degrees 

Number  of  buckets,  52  Number  of  guides,  32 

Inserting  in  the  above  formula  the  values  of  alt  a,  and  a0, 
placing  for  u^  —u2  its  value  (^u:N)2  (rt2  —  r2),  where  N  is  the 
number  of  revolutions  per  minute,  it  reduces  to 


q  =$.44C\2gh  +  0.04287V2 

From  this  the  value  of  c  may  be  computed  for  each  of  the 
seven  experiments  and  the  following  tabulation  shows  the 
results,  the  first  four  columns  giving  the  number  of  the  ex- 
periment, the  observed  head,  number  of  revolutions  per 
minute,  and  discharge  in  cubic  feet  per  second.  The  fifth 
column  gives  the  theoretic  discharge  computed  from  the 
above  formula,  taking  the  coefficient  as  unity,  and  the  last 
column  is  derived  by  dividing  the  observed  discharge  q  by 
the  theoretic  discharge  Q.  The  discrepancy  of  5  or  6  per- 
cent is  smaller  than  might  be  expected,  since  the  formula 
does  not  consider  fractional  resistances. 


No. 

h 

TV 

q 

Q 

c 

21 

17.16 

63-5 

117.01 

123.1 

0.950 

2O 

17.27 

70.0 

118.37 

125.2 

°-94S 

19 

17-33 

75-° 

"9-53 

126.8 

o-943 

18 

17-34 

80.0 

121.15 

128.4 

0.944 

17 

17.21 

86.0 

122.41 

130.0 

0.942 

16 

17.21 

93-2 

124.74 

132-5 

0.941 

i5 

17.19 

IOO.O 

i27-73 

J34-9 

0.947 

A  satisfactory  formula  for  the  discharge  through  a  tur- 
bine when  the  gate  is  partly  depressed  is  difficult  to  deduce, 
because  the  loss  of  head  which  then  results  can  only  be  ex- 
pressed by  the  help  of  experimental  coefficients  similar  to 


ART.  167  THEORY   OF    REACTION   TURBINES  449 

those  given  in  Art.  88  for  the  sliding  gate  in  a  water  pipe 
and  the  values  of  these  for  turbines  are  not  known.  It  is, 
however,  certain  that  for  each  particular  gate  opening  the 
discharge  is  given  by 

q=m\/2gh  +  ul*  —u- 

in  which  m  depends  upon  the  areas  of  the  orifices  and  the 
height  to  which  the  gate  is  raised.  For  instance,  in  the 
tests  of  the  above  Boyden  turbine  the  mean  value  of  m  for 
full  gate  opening  is  3.25,  but  when  the  gate  was  only  six- 
tenths  open  its  value  was  2.81,  and  when  the  gate  was  two- 
tenths  open  its  value  was  1.36.  Each  form  and  size  of  reac- 
tion turbine  has  its  own  values  of  m,  depending  upon  the 
area  of  its  orifices,  and  when  these  have  been  determined  a 
turbine  may  be  used  as  a  water  meter  to  measure  the  dis- 
charge with  a  fair  degree  of  precision. 

Prob.  166a.  Check  the  constants  in  the  above  formula  for 
the  Boyden  turbine,  and  compute  the  values  of  c  for  experi- 
ments 15  and  21. 

Prob.  1666.  Consult  Francis'  Lowell  Hydraulic  Experiments, 
pages  67-75,  and  compute  the  coefficient  m  for  experiments  30 
and  31  on  the  center  vent  Boott  turbine. 


ART.  167.     THEORY  OF  REACTION  TURBINES 

The  theory  of  reaction  turbines  may  be  said  to  include 
two  problems :  first,  given  all  the  dimensions  of  a  turbine  and 
the  head  under  which  it  works,  to  determine  the  maximum 
efficiency,  and  the  corresponding  speed,  discharge,  and 
power;  and  second,  having  given  the  head  and  the  quantity 
of  water,  to  design  a  turbine  of  high  efficiency.  This  article 
deals  only  with  the  first  problem,  and  it  should  be  said  at 
the  outset  that  it  cannot  be  fully  solved  theoretically,  even 
for  the  best-conditioned  wheels,  on  account  of  losses  in  foam, 
friction,  and  leakage.  The  investigation  will  be  limited  to 
the  case  of  full  gate,  since  when  the  gate  is  partially  de^ 


450  TURBINES  CHAP,  xiv 

pressed  a  loss  of  energy  results  from  sudden  expansion. 

The  notation  will  be  the  same  as  that  used  in  Chapters 
XI  and  XIII,  and  as  shown  in  Figs.  1656  and  165<? ;  the  rea- 
soning will  apply  to  both  outward-  and  inward-flow  tur- 
bines. Let  r  be  the  radius  of  the  circumference  where  the 
water  enters  the  wheel  and  rl  that  of  the  circumference 
where  it  leaves,  let  u  and  ul  be  the  corresponding  velocities 
of  revolution ;  then  url  =  up.  Let  v0  be  the  absolute  velocity 
with  which  the  water  leaves  the  guides  and  enters  the  wheel, 
and  V  its  velocity  of  entrance  relative  to  the  wheel ;  let  a  be 
the  approach  angle  and  <£  be  the  entrance  angle  which  these 
velocities  make  with  the  direction  of  u.  At  the  exit  circum- 
ference let  Vi  be  the  relative  velocity  with  which  the  water 
leaves  the  guides,  and  vl  its  absolute  velocity;  let  /?  be  the 
exit  angle  which  V±  makes  with  this  circumference.  Let  a0, 
a,  and  at  be  the  areas  of  the  guide  orifices,  the  entrance,  and 
the  exit  orifices  of  the  wheel,  respectively,  measured  per- 
pendicular to  the  directions  of  v0,  V,  and  Vlf  Let  dQ,  d,  and 
dl  be  the  depths  of  these  orifices;  when  the  gate  is  fully 
raised  dQ  becomes  equal  to  d. 

The  areas  a0,  a,  alt  neglecting  the  thickness  of  the  guides 
and  vanes,  and  taking  the  gate  as  fully  open,  have  the  values 

.  a0  =  2nrd  sin  a:  a  =  2nrd  sin<£        at  =  2nr1d1  sin/? 

and  since  these  areas  are  fully  filled  with  water, 

q=v0.27:rd  sina  =V .znrd  sin<£  =  F1. 27ir1d1  sin/?       (167)! 

These  relations,  together  with  the  formulas  of  the  last  article 
and  the  geometrical  conditions  of  the  parallelograms  of  ve- 
locities, include  the  entire  theory  of  the  reaction  turbine. 

In  order  that  the  efficiency  of  the  turbine  may  be  as  high 
as  possible  the  water  must  enter  tangentially  to  the  vanes, 
and  the  absolute  velocity  of  the  issuing  water  must  be  as 
small  as  possible.  The  first  condition  will  be  fulfilled  when 
n  and  VQ  are  proportional  to  the  sines  of  the  angles  <£  —  a  and 


ART.  167  THEORY   OF    REACTION    TURBINES  451 

(/>.  The  second  will  be  secured  by  making  ul  =  V1  in  the 
parallelogram  at  exit,  as  then  the  diagonal  vl  becomes  very 
small.  Hence 

•±  =  sin(^-a) 
v0          sm<f> 

are  the  two  conditions  which  should  obtain  in  order  that 
the  hydraulic  efficiency  may  be  a  maximum. 

Now  making  Vt  =u^  in  the  third  quantity  of  (167)!  and 
equating  it  to  the  first,  there  results 

ul  _  rd  sina  u      r2d  sina 

==     i    •    ~r\        and  •  ==    ir~j    .    ~n 

VQ     r^  sin/?  ^o     r^d  sm/j 

Also  making  Vl=u1  in  (166)!  and  substituting  for  V2  its 
value  MZ  +  VO?  —  2MVQ  cosa  from  the  triangle  at  A  between  u 
and  v0,  there  is  found  the  important  relation 

uvQcosa=gh  (167), 

which  gives  another  condition  between  u  and  VQ.  The  ve- 
locity v0,  with  which  the  water  enters,  hence  depends  upon 
the  speed  of  the  wheel  as  well  as  upon  the  head  h. 

Thus  three  equations  between  two  unknown  quantities  u 
and  v0  have  been  deduced  for  the  case  of  maximum  hy- 
draulic efficiency,  namely, 

u       sm(<f>  —  a)          u       r"*d  sina  gh 


If  the  values  of  the  velocities  u  and  v0  be  found  from  the 
first  and  third  equations,  they  are 


gh  sin(<£  —  a)  gh 


the  first  of  which  is  the  advantageous  velocity  of  the  circum- 
ference where  the  water  enters,  and  the  second  is  the  abso- 
lute velocity  with  which  the  water  leaves  the  guides  and 
enters  the  wheel.  In  order,  however,  that  these  expressions 


452  TURBINES  CHAP,  xiv 

may  be  correct,  the  first  and  second  values  of  U/VQ  must  also 
be  equal,  and  accordingly 


'r^d,  sin/? 

which  is  the  necessary  relation  between  the  dimensions  and 
angles  of  the  wheel  in  order  that  this  theory  may  apply. 

For  a  turbine  so  constructed  and  running  at  the  advan- 
tageous speed  the  hydraulic  efficiency  is 

v^  2U^  sin2^/? 

€  =  I  --  r  ==  I  —  7  ' 

2gh  gh 

and  substituting  for  u^  its  value  in  terms  of  u  from  (167)4 
and  having  regard  to  (167  )5,  this  becomes 

e  =  i  -  -T  tana  tanj^  (167)« 


The  discharge  under  the  same  conditions  is  q=aQv0,  and 
lastly  the  work  of  the  wheel  per  second  is  k  =  wqhe. 

The  result  of  this  investigation  is  that  the  general  prob- 
lem of  investigating  a  given  turbine  cannot  be  solved  theo- 
retically, unless  it  be  so  built  as  to  approximately  satisfy 
the  condition  in  (167)5.  If  this  be  the  case,  it  may  be  dis- 
cussed by  the  formulas  deduced.  Even  then  no  very  satis- 
factory conclusions  can  be  drawn  from  the  numerical  values, 
since  the  formulas  do  not  take  into  account  the  loss  by 
friction  and  that  of  leakage.  To  determine  the  efficiency, 
best  speed,  and  power  of  a  given  turbine,  the  only  way  is 
to  actually  test  it  by  the  method  described  in  Art.  140.  The 
above  formulas  are,  however,  of  great  value  in  the  discussion 
of  the  design  of  turbines.  More  exact  formulas,  from  a 
theoretical  standpoint,  may  be  derived  by  using  the  con- 
dition V1  =  HI  cos/?  instead  of  Vl  =  u^  to  determine  the  exit 
velocity  vl  (Art.  159),  but  these  are  very  complex  in  form, 
and  numerical  values  computed  from  them  differ  but  little 
from  those  found  from  the  formulas  here  established. 


ART.  168  DESIGN    OF   REACTION   TURBINES  453 

If  the  coefficient  of  discharge  of  a  turbine  be  known  (Art. 
166),  the  advantageous  speed  and  corresponding  discharge 
may  be  closely  computed.  For  this  purpose  the  condition 
ut  =  Vl  =q/al  is  to  be  used.  Inserting  in  this  the  value  of 
q  from  (167)2  and  solving  for  uly  there  is  found 


U2  = 


r.     *  n 


which  gives  the  advantageous  velocity  of  the  circumference 
where  the  water  leaves  the  wheel,  and  then  by  (166)2  the 
discharge  can  be  obtained.  As  an  example,  take  the  case 
of  Holyoke  test  No.  275,  where  rx  =  27^  inches,  r  =  2 1 J  inches, 
^  =  23.8  feet,  a0  =  2.o66,  a  =  5. 526,  ax  =  1.949  square  feet, 
a=252°>  </>=9o°>  /?  =  Iif°-  Assuming  ^=0.95,  as  the 
turbine  is  similar  to  that  investigated  in  the  last  article,  the 
above  formula  gives  u^  =31.24  feet  per  second,  which  corre- 
sponds to  130  revolutions  per  minute,  and  this  agrees  well 
with  the  actual  number  138.  The  efficiency  found  by  the 
test  at  that  speed  was  0.79,  which  is  a  very  much  less  value 
than  the  above  theoretic  formula  gives,  since  this  formula 
was  derived  without  taking  into  account  the  friction  losses 
within  and  without  the  wheel. 

Prob.  167.  For  the  case  of  the  last  problem  r  =  4.67,  ^  =  3.95, 
^=1.01,^  =  1.23,^=13.4  feet,  a  =9°. 5,  </>  =  i  19°, /?=n°.  Com- 
pute the  areas  a0,  a,  alt  and  the  advantageous  speed.  Compute 
also  the  velocity  with  which  the  water  enters  the  wheel. 

ART.  168.      DESIGN  OF  REACTION  TURBINES 

The  design  of  an  outward-  or  inward-flow  turbine  for  a 
given  head  and  discharge  includes  the  determination  of  the 
dimensions  r,  r^  d,  dly  and  the  angles  a,  /?,  and  <j>.  These 
may  be  selected  in  very  many  different  ways,  and  the  for- 
mulas of  the  last  article  furnish  a  guide  how  to  do  this  so  as 
to  secure  a  high  degree  of  efficiency. 


454  TURBINES  CHAP,  xiv 

First,  it  is  seen  from  (167)6  that  the  approach  angle  a 
and  the  exit  angle  /?  should  be  small,  but  that,  as  in  other 
wheels,  /?  has  a  greater  influence  than  a.  However,  /?  must 
usually  be  greater  for  an  inward-flow  than  for  an  outward- 
flow  wheel  in  order  to  make  the  orifices  of  exit  of  sufficient 
size.  For  the  entrance  angle  <j>  a  good  value  is  90  degrees, 
and  in  this  case  the  velocity  u  is  always  that  due  to  one- 
half  the  head,  as  seen  from  (167)  4.  The  radii  r  and  rx  should 
not  differ  too  much,  as  then  the  frictional  resistance  of  the 
flowing  water  and  the  moving  wheel  would  be  large.  It  is 
also  seen  that  the  efficiency  is  increased  by  making  the  exit 
depth  dl  greater  than  the  entrance  depth  d,  but  usually  these 
cannot  greatly  differ,  and  are  often  taken  equal. 

Secondly,  it  is  seen  that  the  dimensions  and  angles  should 
be  such  as  to  satisfy  the  formula  (167)  5,  since  if  this  be  not 
the  case  losses  due  to  impact  at  entrance  will  occur  which 
will  render  the  other  formulas  of  little  value. 

As  a  numerical  illustration  let  it  be  required  to  design  an 
outward-flow  reaction  turbine  which  shall  use  120  cubic  feet 
per  second  under  a  head  of  18  feet  and  make  100  revolutions 
per  minute.  Let  the  entrance  angle  <f>  be  taken  at  90  de- 
grees, then  from  formula  (167)4  the  advantageous  velocity  of 
the  inner  circumference  is 

u=\/  32.16  Xi8  =24.06  feet  per  second, 

and  hence  the  inner  radius  of  the  wheel  is    . 
60X24.06 


271  XlOO 


=2.298  feet. 


Now  let  the  outer  radius  of  the  wheel  be  three  feet,  and  also 
let  the  depths  d  and  d^  be  equal  ;  then  from  (167)  5 

sin/?      /2.298V 
—  -  =   -      -      =0.5866 
tana     \3.ooo/ 

If  the  approach  angle  a  be  taken  as  30  degrees,  the  value  of 
the  exit  angle  ft  to  satisfy  this  equation  is  19°  48',  and  from 


ART.  168  DESIGN    OF    REACTION    TURBINES  455 

(167)  6  the  hydraulic  efficiency  is  0.899.  If,  however,  a  be  24 
degrees,  the  value  of  /?  is  15°  08'  and  the  hydraulic  efficiency 
is  0.941  ;  these  values  of  a  and  /?  will  hence  be  selected. 

The  depth  d  is  to  be  chosen  so  that  the  given  quantity  of 
water  may  pass  out  of  the  guide  orifices  with  the  proper 
velocity.  This  velocity  is,  from  (167)  4, 

v0  =  24.o6/cos  24°  =  26.34  feet  per  second  ; 
and  hence  the  area  of  the  guide  orifices  should  be 

a0  =  i2o/26.34  =4.556  square  feet, 
from  which  the  depth  of  the  orifices  and  wheel  is 

d  =4.556/27^  sin  24°  =0.7  76  feet. 

As  a  check  on  the  computations  the  velocities  V  and  V19 
with  the  corresponding  areas  a  and  a0,  may  be  found,  and  d 
be  again  determined  in  two  ways.  Thus, 

V  =  v0  sin  24°  =  10.71          Fl  =  w1  =  ttr1/r  =  3i.42  ft.  per  sec. 
a=  120/10.71  =  11.204       ax  =  120/31.42  =3.820  square  feet. 

6^  =  3.820/27^  sin/?  =  o.776  feet. 


And  this  completes  the  preliminary  design,  which  should 
now  be  revised  so  that  the  several  areas  may  not  include 
the  thickness  of  the  guides  and  vanes  (Art.  169). 

Although  the  hydraulic  efficiency  of  this  reaction  tur- 
bine is  94  percent,  the  practical  efficiency  will  probably  not 
exceed  80  per  cent.  About  2  percent  of  the  total  work  will 
be  lost  in  axle  friction.  The  losses  due  to  the  friction  of 
the  water  in  passing  through  the  guides  and  vanes,  together 
with  that  of  the  wheel  revolving  in  water,  and  perhaps  also 
a  loss  in  leakage,  will  probably  amount  to  more  than  one- 
tenth  of  the  total  work.  All  of  these  losses  influence  the 
advantageous  velocity,  so  that  a  test  would  be  likely  to 
show  that  the  highest  efficiency  would  obtain  for  a  speed 
somewhat  less  than  100  revolutions  per  minute. 


456  TURBINES  CHAP,  xiv 

Prob.  168.  Design  an  inward-flow  reaction  turbine  which 
shall  use  120  cubic  feet  of  water  per  second  under  a  head  of  i& 
feet  while  making  100  revolutions  per  minute,  taking  <^>  =  68°, 
a  =  10°,  and  /?  =  2i°.  Also  taking  9=75°,  a  =15°,  and  /?=2o°. 


ART.  169.     GUIDES  AND  VANES 

The  discussions  in  the  last  two  articles  have  neglected 
the  thickness  of  the  guides  and  vanes.  As  these,  however, 
occupy  a  considerable  space,  a  more  correct  investigation  will 
here  be  made  to  take  them  into  account.  Let  /  be  the  thick- 
ness of  a  guide  and  n  their  number,  ^  the  thickness  of  a  vane 
and  HI  their  number.'  Then  the  areas  a0,  a,  and  at  perpen- 
dicular to  the  directions  of  v0,  V,  and  Vx  are  strictly 


sina  —nt)d         a 

l  sin/?  —n^t^d^ 


and  the  expressions  for  the  discharge  in  (167)  l  are 

q  =  a0vQ  =aV  =a1F1 
and,  since  Vl  equals  ult  these  give 


also,  the  necessary  condition  in  (167)  5  becomes 

sin  ((ft  —a)  _  a0r 
sin  cf)          a^i 

and  the  greatest  hydraulic  efficiency  of  the  turbine  when 
running  at  the  advantageous  speed  is  given  by 

r^ 


r2       sin<         cosa 


in  which,  of  course,  sin  (<f>  —  a)  /sin  <£  may  be  replaced  by  its 
equivalent  aQr/a^.  The  advantageous  speed  is,  as  before,, 
given  by  (167>., 


ART.  169  GUIDES   AND   VANES  457 

To  discuss  a  special  case,  let  the  example  of  the  last 
article  be  again  taken.  An  outward-flow  turbine  is  to  be 
designed  to  use  120  cubic  feet  of  water  under  a  head  of  18 
feet  while  making  100  revolutions  per  minute,  the  gate  being 
fully  opened.  The  preliminary  design  has  furnished  the 
values  r  =  2.2gS  feet,  ^=3.000  feet,  d=dl=o.^6  feet, 
<£=9o°,  a  =24°,  P  =  i$°  08'.  It  is  now  required  to  revise 
these  so  that  24  guides  and  36  vanes  may  be  introduced. 
Each  of  these  will  be  made  one-half  an  inch  thick,  but  on 
the  inner  circumference  of  the  wheel  the  vanes  will  be 
thinned  or  rounded  so  as  to  prevent  shock  and  foam  that 
might  be  caused  by  the  entering  water  impinging  against 
their  ends  (see  Fig.  1730).  If  the  radii  and  angles  remain 
unchanged,  the  effect  of  the  vanes  will  be  to  increase  the 
depth  of  the  wheel,  which  is  now  0.702  feet  wide  and  0.776 
feet  deep.  As  these  are  good  proportions,  it  will  perhaps 
be  best  to  keep  the  depth  and  the  radii  unchanged,  and  to 
see  how  the  angles  and  the  efficiency  will  be  affected. 

Since  the  vanes  are  to  be  thinned  at  the  inner  circumfer- 
ence, the  area  a  is  unaltered  and  its  value  is  simply  znrd  sin<£. 
Hence  ^  remains  90  degrees,  and  V  is  unchanged.  This 
requires  that  the  area  a  should  remain  the  same  as  before. 
The  area  a^  is  also  the  same,  as  its  value  is  q/u±.  Accord- 
ingly the  equations  result 

4.556  =(27rr  sina  —  24*)^        3. 820  =  (27^  sin/? —36^)^ 

in  which  a  and  /?  are  alone  unknown.  Inserting  the  nu- 
merical values  and  solving,  a  =28°  26'  and  /?  =  19°  55',  both 
being  increased  by  about  4^  degrees.  The  efficiency  is 
now  found  to  be  0.898,  a  decrease  of  0.043,  due  to  the  intro- 
duction of  the  guides  and  vanes. 

The  efficiency  may  be  slightly  raised  by  making  the  outer 
depth  dl  greater  than  the  inner  depth  d.  For  instance,  let 
dl  =0.816,  while  d  remains  0.776;  then  /?  is  found  to  be  19° 
06',  and  0=0.906.  But  another  way  is  to  thin  down  the 


458  TURBINES  CHAP,  xiv 

vanes  at  the  exit  circumference  and  thus  maintain  the  full 
area  ax  with  a  small  angle  ft.  If  this  be  done  in  the  present 
case  d^  may  be  kept  at  0.776  feet,  ft  be  reduced  to  about  16 
degrees,  and  the  efficiency  will  then  be  about  0.92  or  0.93. 

No  particular  curve  for  the  guides  and  vanes  is  required, 
but  it  must  be  such  as  to  be  tangent  to  the  circumferences 
at  the  designated  angles.  The  area  between  two  vanes  on 
any  cross-section  normal  to  the  direction  of  the  velocity 
should  also  not  be  greater  than  the  area  at  entrance;  in 
order  to  secure  this  vanes  are  frequently  made  much  thicker 
at  the  middle  than  at  the  ends  (see  Fig.  17  3e). 

Prob.  169a.  Find  the  advantageous  speed  and  the  probable 
discharge  and  power  of  the  turbine  designed  above  when  Bunder 
a  head  of  50  feet. 

Prob.  1696.  Revise  the  design  of  Prob.  168  by  finding  the 
influence  of  16  guides  and  12  vanes  upon  the  radii  of  the  cir- 
cumferences and  the  depth  of  the  wheel. 

ART.  170.     DOWNWARD-FLOW  TURBINES 

Downward-  or  parallel-flow  turbines  are  those  in  which 
the  water  passes  through  the  wheel  without  changing  its 


FIG.  170a 

distance  from  the  axis  of  revolution.  In  Fig.  170a  is  a  semi- 
vertical  section  of  the  guide  and  wheel  passages,  and  also  a 
development  of  a  portion  of  a  cylindrical  section  showing 
the  inner  arrangement.  The  formula  for  the  discharge  can 
be  adapted  to  this  by  making  u±  =  u.  In  this  turbine  there 
is  no  action  of  centrifugal  force,  so  that  the  relative  exit 


ART.  170 


DOWNWARD-FLOW  TURBINES 


459 


velocity  Vx  is  equal  to  the  relative  entrance  velocity  V. 

The  great  advantage  of  this  form  of  turbine  is  that  it  can 
be  set  some  distance  above  the  tail  race  and  still  obtain  the 
power  due  to  the  total  fall.  This  distance  cannot  exceed  34 
feet,  the  height  of  the  water 
barometer,  and  usually  it  does 
not  exceed  25  feet.  Fig.  1706 
shows  in  a  diagrammatic  way  a 
cross-section  of  the  penstock  P, 
the  guide  passages  G,  the  wheel 
W,  and  the  air-tight  draft  tube 
T,  from  which  the  water  escapes 
by  a  gate  E  to  the  tail  race.  The 
pressure-head  H^  on  the  exit  ori- 
fice is  here  negative,  so  that  the 
air  pressure  equivalent  to  this 
head  is  added  to  the  water  pres- 
sure in  the  penstock,  and  hence 
the  discharge  through  the  guides 
occurs  as  if  the  wheel  were  set  at 
the  level  of  the  tail  race.  Strict- 
ly speaking  a  vacuum,  more  or 
less  complete,  is  formed  just  be- 
low the  wheel  into  which  the  water  drops  with  a  low  abso- 
lute velocity,  having  surrendered  to  the  wheel  nearly  all  its 
energy.  Draft  tubes  are  also  often  used  with  inward-flow 
turbines  when  these  are  set  above  the  tail  race. 

Let  h  be  the  total  head  between  the  water  levels  in  the 
head  and  tail  races,  h0  the  depth  of  the  entrance  orifices  of 
the  wheel  below  the  upper  level,  hl  the  vertical  height  of  the 
wheel,  and  h2  the  height  of  the  exit  orifices  above  the  tail 
race;  so  that  h  =/*0  +  /£1  +  ^2.  Let  H  and  Ht  be  the  heads 
which  measure  the  absolute  pressures  at  the  entrance  and 
exit  orifice  of  the  wheel,  and  ha  the  height  of  the  water  ba- 
rometer. Let  v0  be  the  absolute  velocity  with  which  the 


FIG.  1706 


460  TURBINES  CHAP,  xiv 

water  leaves  the  guides  and  enters  the  vanes,  and  V  and  Vl 
the  relative  velocities  at  entrance  and  exit.  Then  from  the 
theorem  of  energy  in  steady  flow  (Art.  32), 


Adding  these  two  equations  there  results 

v0*-V*  +  V1*  =  2g(h0+h1 
But  ha  —Hi  is  equal  to  h2,  and  hence 


This  formula  is  the  same  as  (166)i  if  u  be  made  equal  to  ult 
and  >hence  all  the  formulas  of  the  last  three  articles  apply  to 
the  downward-flow  reaction  turbine  by  making  equal  the 
velocities  u  and  uv  as  also  the  radii  r  and  rr 

Let  r  be  the  mean  radius  and  u  the  mean  velocity  of  the 
entrance  and  exit  orifices  of  the  wheel,  let  d  be  the  width  of 
-the  entrance  orifices  and  dl  that  of  the  exit  orifices.  Let  a 
be  the  approach  angle  which  the  direction  of  the  entering 
water  makes  with  that  of  the  velocity  u,  or  the  angle  which 
the  guides  make.  with  the  upper  plane  of  the  wheel  (Fig. 
170a)  ;  let  <f>  be  the  entrance  angle  which  the  vanes  make 
with  that  plane,  and  /?  the  acute  exit  angle  which  they  make 
with  the  lower  plane.  Then  the  values  of  the  advantageous 
velocity  u  and  the  entering  velocity  v0  are 


u  = 


I 

\< 


cosa  sin(0  —  a) 

and  the  necessary  relation  between  the  angles  of  the  vanes 
and  the  dimensions  of  the  wheel  is 

sin(<£  —  a)  _  d  sma      a0 
sin0  d^  sin/?  ~  at 

while  the  hydraulic  efficiency  of  the  turbine  is 

a0  sinH/?  d 

0=1-2-      — ^  =!i  -  -T- tana  tan-|/? 
ax   cosa  at 

To  these  equations  is  to  be  added  the  condition  that  the 
pressure-head  Ht  cannot  be  less  than  that  of  a  vacuum,  and 


ART.  171  IMPULSE  TURBINES  461 

on  account  of  air  leakage  it  must  be  practically  greater  ;  thus 

Hl  >  o         and         h2  <  ha 

that  is,  the  height  of  the  wheel  orifices  above  the  tail  race 
must  be  less  than  the  height  of  the  water  barometer. 

As  an  example  of  design,  let  <f>  =90°  and  a  =30°.  Then 
u  =  \/gh,  or  the  velocity  due  to  one-half  the  head;  and 
vo  =  \/%gh,  or  a  velocity  due  to  two-thirds  of  the  head. 
Prom  the  above  formulas,  taking  dl=%d,  the  value  of  /?  is 
22°  38'  and  the  efficiency  is  found  to  be  0.92.  This  value 
will  be  lowered  by  the  introduction  of  guides  and  vanes,  as 
well  as  by  friction,  so  that  perhaps  not  more  than  0.80  will 
"be  obtained  in  practice. 

Prob.  170a.  A  downward-flow  turbine  has  d  =  d^  a  =  i6°, 
^=15°,  h  =  $o  feet;  compute  the  angle  <£,  the  best  speed,  and 
the  hydraulic  efficiency. 

Prob.  1706.  A  downward-flow  turbine  with  draft  tube  has 
its  exit  orifices  7.5  feet  above  the  level  of  the  tail  race,  and  it 
uses  87  cubic  feet  of  water  per  second  under  a  head  of  25  feet. 
What  horse-power  will  this  turbine  deliver  if  its  efficiency,  as 
measured  by  the  friction  brake,  is  76  percent? 

ART.  171.      IMPULSE  TURBINES 

Whenever  a  turbine  is  so  arranged  that  the  channels  be- 
tween the  vanes  are  not  fully  filled  with  water,  it  ceases  to 
act  as  a  reaction  turbine  and  becomes  an  impulse  turbine. 
A  turbine  set  above  the  level  of  the  tail  race  becomes  an  im- 
pulse turbine  when  the  gate  is  partially  lowered,  unless  the 
gates  are  arranged  over  the  exit  orifices. 

The  velocity  with  which  the  water  leaves  the  guides  in 
an  impulse  turbine  is  simply  \/2gh0,  where  hQ  is  the  head  on 
the  guide  orifices.  The  rules  and  formulas  in  Art.  159  apply 
in  all  respects,  and  for  a  well-designed  wheel  the  entrance 
angle  <£  is  double  the  approach  angle  a,  the  advantageous 
speed  and  corresponding  hydraulic  efficiency  are 


A 


2  cos2a  r  cosa 


462  TURBINES  CHAP,  xiv 

while  the  discharge  is  q  =a0V2gh0  and  the  work  of  the  tur- 
bine per  second  is  k  =  wqh0e. 

As  an  example,  suppose  that  the  reaction  turbine  de- 
signed in  Art.  168  were  to  act  as  an  impulse  turbine,  and 
the  angles  a  and  /?  remaining  at  24°  and  15°  08',  the  radii  r 
and  r1  being  2.298  and  3.000  feet.  It  would  then  be  neces- 
sary that  (j>  should  be  48°  instead  of  90°  in  order  to  secure 
the  best  results.  Under  a  head  of  18  feet  the  velocity  of 
flow  from  the  guides  would  be  34.02  feet  per  second  instead 
of  26.34.  The  velocity  of  the  inner  circumference  would  be 
18.63  feet  per  second  instead  of  24.06,  so  that  the  number 
of  revolutions  per  minute  would  be  about  77  instead  of  100. 
The  efficiency  would  be  0.96,  or  almost  exactly  the  same 
as  before.  If,  however,  the  angle  <£  were  to  remain  90°, 
the  efficiency  would  be  materially  lowered,  since  then  the 
water  could  not  enter  tangentially  to  the  vanes  and  a  loss 
in  impact  would  necessarily  result. 

Impulse  turbines  revolve  slower  than  reaction  turbines 
under  the  same  head,  but  the  relative  entrance  velocity  V 
is  greater,  and  hence  more  energy  is  liable  to  be  spent  in 
shock  and  foam.  In  impulse  turbines  the  entrance  angle 
<j>  should  be  double  the  approach  angle  a,  but  in  reaction 
turbines  it  is  often  greater  than  30:,  and  its  value  depends 
upon  the  exit  angle  /?;  hence  the  vanes  in  impulse  turbines 
are  of  sharper  curvature  for  the  same  values  of  a  and  /?.  In 
impulse  turbines  the  efficiency  is  not  lowered  by  a  partial 
closing  of  the  gates,  whereas  the  sudden  enlargement  of  sec- 
tion causes  a  material  loss  in  reaction  turbines.  The  advan- 
tageous speed  of  an  impulse  turbine  remains  the  same  for 
all  positions  of  the  gate,  but  with  a  reaction  turbine  it  is 
very  much  slower  at  part  gate  than  at  full  gate.  For  many 
kinds  of  machinery  it  is  important  to  maintain  a  constant 
speed  for  different  amounts  of  power,  and  with  a  reaction 
turbine  this  can  only  be  done  by  a  great  loss  in  efficiency. 
When  the  water  supply  is  low  the  impulse  turbine  hence  has 
a  marked  advantage  in  efficiency.  A  further  merit  of  the 


ART.  172 


SPECIAL  DEVICES 


463 


impulse  turbine  is  that  it  may  be  arranged  so  that  water 
enters  only  through  a  part  of  the  guides,  while  this  is  impos- 
sible in  reaction  turbines.  On  the  other  hand,  reaction  tur- 
bines can  be  set  below  the  level  of  the  tail  race  or  above  it, 
using  a  draft  tube  in  the  latter"  case,  and  still  secure  the 
power  due  to  the  total  fall,  whereas  an  impulse  turbine  must 
always  be  set  above  the  tail-race  level  and  loses  all  the  fall 
between  that  level  and  the  guide  orifices. 

Prob.  171a.  Compare  the  advantageous  speeds  of  impulse 
and  reaction  turbines  when  the  velocity  of  the  water  issuing  from 
the  guide  orifices  is  the  same. 

Prob.  1716.  Design  an  outward-flow  impulse  turbine  which 
shall  use  120  cubic  feet  of  water  per  second  under  a  head  of 
1 8  feet  and  make  100  revolutions  per  minute.  Compare  the 
dimensions  and  angles  with  those  of  the  reaction  turbine  de- 
signed for  the  same  data  in  Art.  168. 

ART.  172.      SPECIAL  DEVICES 

Many  devices  to  increase  the  efficiency  of  reaction  tur- 
bines, particularly  at  part  gate,  have  been  proposed.  In 
the  Fourneyron  turbine  a  common  plan  is  to  divide  the 
wheel  into  three  parts  by  horizontal  partitions  between  the 
vanes  so  that  these  are  completely  filled  with  water  when 
the  gate  is  either  one-third  or  two-thirds  closed  (see  Fig. 
17 3d).  The  surface  exposed  to  friction  is  thus,  however, 
materially  increased  at  full  gate. 

The  Boyden  diffuser  is  another  device  used  with  out- 
ward-flow reaction  turbines.  This  consists  of  a  fixed  wooden 
annular  frame  D  placed  around 
the  wheel  W,  through  which  the 
water  must  pass  after  exit  from 
the  wheel.  Its  width  is  about 
four  or  five  times  that  of  the 
wheel,  and  at  the  outer  end  its 
depth  becomes  about  double  that  FIG.  172 

of  the  wheel.     The  effect  of  this  is  like  a  draft  tube,  and  al- 


464  TURBINES  CHAP,  xiv 

though  the  absolute  velocity  of  the  water  when  .ssuing  from 
the  wheel  is  greater  than  before,  the  absolute  velocity  of 
the  water  coming  out  of  the  difluser  is  less,  and  hence  a 
greater  amount  of  energy  is  imparted  to  the  turbine.  It  has 
been  shown  above  that  the  efficiency  of  a  reaction  turbine 
is  increased  by  making  the  exit  depth  dt  greater  than  the 
entrance  depth  d,  and  the  fixed  diffuser  produces  the  same 
result.  By  the  use  of  this  diffuser  Boy  den  increased  the  ef- 
ficiency of  the  Fourneyron  reaction  turbine  several  percent. 

The  pneumatic  turbine  of  Girard  was  devised  to  over- 
come the  loss  in  reaction  turbines  due  to  a  partial  closing  of 
the  gate.  The  turbine  was  enclosed  in  a  kind  of  bell  into 
which  air  could  be  pumped,  thus  lowering  the  tail- water 
level  around  the  wheel.  At  part  gate  this  pump  is  put  into 
action,  and  as  a  consequence  the  air  is  admitted  into  the 
wheel,  and  the  water  flowing  through  it  does  not  fill  the 
spaces  between  the  vanes.  Hence  the  action  becomes  like 
that  of  an  impulse  turbine,  and  the  full  efficiency  is  main- 
tained. A  wheel  thus  arranged  should  properly  have  the 
entrance  angle  (f>  double  the  approach  angle  a  in  order  that 
the  advantageous  speed  may  be  always  the  same. 

Turbines  without  guides  have  been  used.  Here  the  ap- 
proach angle  a  is  probably  about  90  degrees,  as  the  water 
would  probably  approach  the  wheel  by  the  shortest  path. 
The  entrance  angle  <j>  would  then  be  made  greater  than  90 
degrees,  and  the  reliance  for  high  efficiency  must  be  upon  a 
small  value  of  the  exit  angle  /?.  But  as  this  can  scarcely  be 
made  smaller  than  15  degrees,  the  hydraulic  efficiency  will 
rarely  exceed  80  percent,  which  by  friction  and  foam  will  in 
practice  be  reduced  to  about  65  percent. 

The  screw  turbine  consists  of  one  or  two  turns  of  a  heli- 
coidal  surface  around  a  vertical  shaft,  the  screw  being  en- 
closed in  a  cylindrical  case.  At  a  point  of  entrance  the 
downward  pressure  of  the  water  can  be  resolved  into  two 
components,  a  relative  velocity  V  parallel  to  the  surface  and 


ART.  173  THE   NIAGARA   TURBINES  465 

a  horizontal  velocity  u  which  corresponds  to  the  velocity  of 
the  wheel.  At  the  point  of  exit  it  can  be  resolved  in  like 
manner  into  Ft  and  ulm  But,  as  in  other  cases,  the  condi- 
tion for  high  efficiency  is  u1  =  Vl,  and  since  the  water  moves 
parallel  to  the  axis,  u^  =u.  Applying  the  general  formulas 
of  Art.  166,  it  is  seen  that  this  can  only  occur  when  the  head 
h  is  zero  or  when  the  velocity  u  is  infinite.  The  screw  tur- 
bine is  hence  like  a  reaction  wheel,  and  high  efficiency  can 
never  practically  be  obtained. 

Prob.  172a.  Consult  Riihlmann's  Maschinenlehre,  vol.  i, 
pp.  360-425,  and  describe  a  scheme  for  "ventilating"  a  tur- 
bine in  order  to  increase  its  efficiency. 

Prob.  1726.  Consult  Weisbach's  Mechanics  of  Engineering, 
vol.  ii  (Du  Bois'  translation),  and  make  sketches  of  a  rotary 
water-pressure  engine.  Show  that  its  action  depends  on  static 
pressure  only,  and  that  it  cannot  be  considered  as  either  an  im- 
pulse or  a  reaction  turbine. 

ART.  173.     THE  NIAGARA  TURBINES 

A  number  of  turbines  have  been  installed  at  Niagara 
Falls,  N.  Y.,  for  the  utilization  of  a  portion  of  the  power  of 
the  great  falls.  Those  to  be  here  briefly  described  are  the 
ten  large  wheels  designed  by  Faesch  and  Picard,  of  Geneva, 
Switzerland,  and  erected  from  1894  to  1900  for  the  Niagara 
Falls  Power  Company.  The  entire  plant  is  to  include 
twenty-one  twin  outward-flow  reaction  turbines,  each  of 
about  5000  horse-power.  It  is  located  about  i£  miles  above 
the  American  fall,  where  a  canal  leads  water  from  the  river 
to  the  wheel  pit.  The  water  is  carried  down  the  pit  through 
steel  penstocks  to  the  turbines,  which  are  placed  136  feet 
below  the  water  level  in  the  canal.  After  passing  through 
the  wheels  the  waste  water  is  conveyed  to  the  river  below 
the  American  fall  by  a  tunnel  7000  feet  long.* 

Fig.  173a  shows  a  cross-section  of  the  wheel  pit,  with  an 
end  view  of  a  penstock,  wheel  case,  and  shaft.  Fig.  1736 

*  Engineering  News,  1892,  vol.  27,  p.  74,  and  1893,  vol.  29,  p.  294. 


466 


TURBINES 


CHAP.  XIV 


exhibits  part  of  a  longi- 
tudinal section  of  the  wheel 
pit  and  a  side  view  of  two 
of  the  penstocks,  with  the  * 
enclosing  cases  and  shafts  S| 
of  the  turbines.  These 
figures  show  a  rock-surface 
wheel  pit,  but  this  surface 
was  later  protected  by  a 
brick  lining  having  a  thick- 
ness of  about  15  inches. 
The  width  of  the  wheel  pit 
is  20  feet  at  the  top  and  16 
feet  at  the  bottom,  and  the 
cylindrical  penstock  is  7^ 
feet  in  diameter.  The  shaft 
of  the  turbine  is  a  steel  tube 
38  inches  in  diameter,  built 
in  three  sections,  and  con- 
nected by  short  solid  steel 
shafts  1 1  inches  in  diameter 
which  revolve  in  bearings. 
At  the  top  of  each  shaft  is 
a  dynamo  for  generating 
the  electric  power. 

In  Fig.  17 3c  is  shown  a 
vertical  section  of  the  lower 
part  of  the  penstock,  shaft, 
and  twin  wheels.  The  water 
fills  the  casing  around  the 
shaft,  passes  both  upward 
and  downward  to  the  guide 
passages,  marked  £,  through 
which  it  enters  the  two 
wheels,  causes  them  to  re- 
volve; and  then  drops  down 


FIGt 


ART.  173 


THE  NIAGARA  TURBINES 


467 


FIG.  1736 


468 


TURBINES 


CHAP.  XIV 


to  the  tail  race  at  the  entrance  to  the  tunnel,  which  carries 
it  away  to  the  river.  The  gate  for  regulating  the  discharge 
is  seen  upon  the  outside  of  the  wheels. 


FIG.  173c 

Fig.  173d  gives  a  larger  vertical  section  of  the  lower  wheel 
with  the  guides,  shaft,  and  connecting  members.  The  guide 
passages,  marked  G,  and  the  wheel  passages,  marked  W,  are 
triple,  so  that  the  latter  may  be  filled  not  only  at  full  gate, 
but  also  when  it  is  one-third  or  two-thirds  opened,  thus 
avoiding  the  loss  of  energy  due  to  sudden  enlargement  of  the 
flowing  stream.  The  two  horizontal  partitions  in  the  wheel 
are  also  advantageous  in  strengthening  it.  The  inner  radius 
of  the  wheel  is  31  \  inches  and  the  outer  radius  is  37^  inches, 
while  the  depth  is  about  1 2  inches.  In  this  figure  the  gates 
are  represented  as  closed. 

In  Fig.  1730  is  shown  a  half-plan  of  one  of  the  wheels,  on 
a  part  of  which  are  seen  the  guides  and  vanes,  there  being  36 


ART.  173 


THE  NIAGARA  TURBINES 


469 


of  the  former  and  32  of  the  latter.     The  value  of  the  ap- 
proach angle  a  is  19°  06',  the  mean  value  of  the  entrance 


STEEL 
AST-STEEL, 

BRONZE 


FIG.  173d 


angle  <£  is  110°  40',  and  the  exit  angle  /?  is  13°  17^'.     Al- 
though the  water  on  leaving  the  wheel  is  discharged  into  the 


FIG. 


air,  the  very  small  annular  space  between  the  guides  and 
vanes,  together  with  the  decreasing  area  between  the  vanes 


470  TURBINES  CHAP,  xiv 

from  the  entrance  to  the  exit  orifices,  ensures  that  the 
,wheels  act  like  reaction  turbines  for  the  three  positions 
of  the  gates  corresponding  to  the  three  horizontal  stages. 

The  average  discharge  through  one  of  these  twin  tur- 
bines is  about  430  cubic  feet  per  second,  and  the  theoretic 
power  due  to  this  discharge  is  6645  horse-powers.  Hence 
if  5000  horse-powers  be  utilized  the  efficiency  is  75.2  percent. 
Under  this  discharge  the  mean  velocity  in  the  penstock  is 
nearly  10  feet  per  second,  but  the  loss  of  head  due  to  friction 
in  the  penstock  will  be  but  a  small  fraction  of  a  foot.  The 
pressure-head  in  the  wheel  case  is  then  practically  that  due 
to  the  actual  static  head,  or  closely  141 J  feet  upon  the  lower 
and  130  feet  upon  the  upper  wheel.  Although  the  penstock 
is  smaller  in  section  than  generally  thought  necessary  for 
such  a  large  discharge,  the  loss  of  head  that  occurs  in  it  is 
insignificant ;  and  it  will  be  seen  in  Fig.  173a  to  be  connected 
with  the  head  canal  and  with  the  wheel  case  by  easy  curves, 
and  that  its  section  is  enlarged  in  making  these  approaches. 

A  test  of  one  of  these  wheels,  made  in  1895,  showed  that 
5498  electrical  horse-powers  were  generated  by  an  expendi- 
ture of  447.2  cubic  feet  of  water  per  second  under  a  head  of 
135.1.  The  efficiency  of  the  dynamo  being  97  percent,  the 
efficiency  of  the  wheel  and  approaches  was  82 J  percent. 
The  water  was  measured,  when  entering  the  penstock,  by  a 
current  meter  of  the  kind  illustrated  in  Art.  40. 

From  formula  (167)4  the  advantageous  velocity  of  the 
inner  circumference  of  the  upper  wheel,  taking  h  =  130^  feet, 
is  found  to  be  68.88  feet  per  second,  and  that  for  the  lower 
wheel,  taking  ^  =  141^  feet,  is  found  to  be  71.73  feet  per 
second.  Perhaps  the  mean  of  these,  or  70.3 1  feet  per  second, 
closely  corresponds  with  the  advantageous  velocity  for  the 
two  combined.  The  number  of  revolutions  per  minute  for 
the  condition  of  maximum  efficiency  is  then  closely  250. 
The  absolute  velocity  of  the  water  when  entering  the  wheel 
is  about  66  feet  per  second,  so  that  the  pressure-head  in  the 


ART.  173  THE   NIAGARA   TURBINES  471 

guide  passages  of  the  upper  wheel  is  nearly  66  feet.  The 
mean  absolute  velocity  of  the  water  when  leaving  the  wheels 
is  about  19  feet  per  second,  so  that  the  loss  due  to  this  is  only 
about  4  percent  of  the  total  head. 

The  weight  of  the  dynamo,  shaft,  and  turbine  is  balanced, 
when  the  wheels  are  in  motion,  by  the  upward  pressure  of 
the  water  in  the  wheel  case  on  a  piston  placed  above  the 
upper  wheel.  The  upper  disk  containing  the  guides  is,  for 
this  purpose,  perforated,  so  that  the  water  pressure  can  be 
transmitted  through  it.  In  Fig.  17 3c  these  perforations  can 
"be  seen,  and  the  balancing  piston  is  marked  B.  The  lower 
disk,  on  the  other  hand,  is  solid,  and  the  weight  of  the  water 
upon  it  is  carried  by  inclined  rods  upward  to  the  wheel  case, 
which  together  with  the  penstock  is  supported  upon  several 
girders.  At  the  upper  end  of  the  shaft  is  a  thrust  bearing 
to  receive  the  excess  of  vertical  pressure,  which  may  be 
either  upward  or  downward  under  different  conditions  of 
power  and  speed. 

A  governor  is  provided  for  the  regulation  of  the  speed, 
and  this  is  located  on  the  surface  near  the  dynamo.  It  is  of 
the  centrifugal-ball  type,  and  so  connected  with  the  main 
shaft  and  the  turbine  gates  that  the  latter  are  partially 
closed  whenever  from  any  cause  the  speed  increases.  These 
gates  are  so  set  that  the  orifices  of  the  upper  and  lower 
wheels  are  not  simultaneously  closed,  one  gate  being  in  ad- 
vance of  the  other  by  about  the  width  of  one  division 
stage.  The .  revolving  field  magnets  of  the  dynamo  also 
serve  as  a  fly-wheel  for  equalizing  the  speed.  With  this 
method  of  regulation  it  is  ensured  that  the  speed  cannot 
increase  more  than  3  or  4  percent  when  25  percent  of  the 
work  is  suddenly  removed. 

The  above  description  refers  to  the  ten  turbines  in  wheel 
pit  No.  1.  The  illustrations  are  those  of  the  wheels  called 
-units  1,  2,  and  3  which  were  installed  in  1894  and  1895. 
Units  4  to  10  inclusive,  installed  in  1898-1900,  are  of  the 


472  TURBINES  CHAP,  xiv 

same  type  except  that  both  the  penstock  and  wheel  case 
have  cast-iron  ribs  on  their  sides  which  rest  on  massive  cast- 
ings built  into  the  masonry  of  the  side  walls.  This  arrange- 
ment dispenses  with  the  supporting  girders  shown  in  Figs. 
173a-173£,  and  gives  much  greater  rigidity  to  both  penstocks 
and  wheels. 

The  excavation  of  a  new  wheel  pit,  called  No.  2,  was 
begun  in  1896,  and  the  installation  of  units  11-21  was  com- 
pleted in  1903.  These  wheels  have  penstocks  and  shafting 
similar  to  those  of  units  1-10,  but  the  wheels  are  of  the  Jou- 
val  type,  the  flow  being  inward  and  downward.  The  wheel 
case  has  the  form  of  a  flattened  sphere,  the  water  entering 
from  one  side  and  passing  through  the  guides  to  a  single 
turbine  64  inches  in  diameter  and  23.5  inches  deep.  After 
leaving  the  wheel,  the  water  passes  to  two  draft  tubes,  each 
about  58  inches  in  diameter,  and  is  discharged  near  the 
in  vert  of  the  tail  race  at  an  angle  of  45°  to  the  horizontal 
axis  of  the  wheel  pit.  The  wheel  case  is  supported  on  these 
two  draft  tubes  as  on  two  legs,  while  the  penstock  is  sup- 
ported on  iron  lugs  in  the  same  way  as  those  of  units 
4-10.  By  these  draft  tubes  the  head  on  the  wheel  is  in- 
creased to  144  feet,  this  being  the  difference  from  the  water 
level  in  the  head  race  to  that  in  the  tail  race.  The  bal- 
ancing pistons  are  below  the  wheels,  and  are  supported  from 
an  independent  pipe  instead  of  from  the  penstock.  Each 
shaft  is  also  supplied  with  an  oil  step-bearing  which  is  de- 
signed to  support,  if  necessary,  the  entire  revolving  weight 
at  the  normal  speed  of  250  revolutions  per  minute. 

Prob.  173a.  Compute  the  hydraulic  efficiency  of  the  tur- 
bines described  above.  Compute  the  velocity  v0  with  which  the 
water  enters  the  lower  wheel  and  the  velocity  v^  with  which  it 
leaves  the  same  when  the  speed  is  250  revolutions  per  minute. 

Prob.  1736.  Compute  the  efficiency  of  a  reaction  wheel  under 
a  head  of  3.5  meters  when  the  radius  of  the  exit  orifices  is 
0.64  meters,  the  coefficient  of  velocity  0.95,  and  the  number 
of  revolutions  per  minute  is  130. 


ART.  173  THE    NIAGARA   TURBINES  473 

Prob.  17 '3c.  Compute  the  discharge  through  a  reaction  tur- 
bine for  which  ^==0.64  and  r=i.oo  meters,  ^  =  0.32,  #  =  1.26, 
a0  =  o.47  square  meters,  when  the  coefficient  of  discharge  is 
0.95  and  the  wheel  makes  220  revolutions  per  minute. 

Prob.  17 3d.  Design  an  outward-flow  reaction  turbine  which 
shall  use  8  cubic  meters  of  water  per  second  under  a  head  of 
12.4  meters,  taking  the  entrance  angle  <j>  as  90  degrees. 

Prob.  173e.  A  dynamo  delivering  4100  kilowatts  has  an 
efficiency  of  97.5  percent,  while  the  efficiency  of  the  turbine  is 
81.3  percent  and  that  of  the  approaches  to  the  turbine  is  99.7 
percent.  The  turbine  is  of  the  Jouval  type,  and  the  difference 
between  the  levels  of  head  and  tail  race  is  14.4  meters.  How 
many  cubic  meters  of  water  are  used  per  second? 


474  NAVAL  HYDROMECHANICS  CHAP,  xv 


CHAPTER  XV 
NAVAL  HYDROMECHANICS 

ART.  174.     GENERAL  PRINCIPLES 

In  this  chapter  is  to  be  discussed  in  a  brief  and  elementary 
manner  the  subject  of  the  resistance  of  water  to  the  motion 
of  vessels,  and  the  general  hydrodynamic  principles  relating 
to  their  propulsion.  The  water  may  be  at  rest  and  the  ves- 
sel in  motion — or  both  may  be  in  motion,  as  in  the  case  of  a 
boat  going  up  or  down  a  river.  In  either  event  the  velocity 
of  the  vessel  relative  to  the  water  need  only  be  considered, 
and  this  will  be  called  v.  The  simplest  method  of  propul- 
sion is  by  the  oar  or  paddle;  then  come  the  paddle  wheel, 
and  the  jet  and  screw  propellers.  The  action  of  the  wind 
upon  sails  will  not  be  here  discussed,  as  it  is  outside  of  the 
scope  of  this  book. 

The  unit  of  measure  used  on  the  ocean  is  generally  the 
nautical  mile  or  knot,  which  is  about  6080  feet,  so  that  knots 
per  hour  may  be  transformed  into  feet  per  second  by  multi- 
plying by  1.69,  and  feet  per  second  may  be  transformed  into 
knots  per  hour  by  multiplying  by  0.592.  On  rivers  the 
speed  is  estimated  in  statute  miles  per  hour,  and  the  corre- 
sponding multipliers  will  be  1.47  and  0.682.  One  kilometer 
per  hour  equals  0.621  miles  per  hour  or  0.91  feet  per  second. 
On  the  ocean  the  weight  of  a  cubic  foot  of  water  is  to  be 
taken  as  about  64  pounds  (it  is  often  used  as  64. 3  2  pounds, 
so  that  the  numerical  value  is  the  same  as  2g) ,  and  in  rivers 
at  62.5  pounds. 

The  speed  of  a  ship  at  sea  is  roughly  measured  by  obser- 
vations with  the  log,  which  is  a  triangular  piece  of  wood 


ART.  174  GENERAL    PRINCIPLES  475 

attached  to  a  cord  which  is  divided  by  tags  into  lengths  of 
about  50 J  feet.  The  log  being  thrown  into  the  water,  it  re- 
mains stationary,  the  ship  moves  away  from  it,  and  the 
number  of  tags  run  out  in  half  a  minute  is  counted;  this 
number  is  the  same  as  the  number  of  knots  per  hour  at 
which  the  ship  is  moving,  since  50!  feet  is  the  same  part  of 
a  knot  that  a  half  minute  is  of  an  hour.  The  patent  log, 
which  is  a  small  self-recording  current  meter,  drawn  in  the 
water  behind  the  ship,  is  however  now  generally  used, 
this  being  rated  at  intervals  (Art.  40).  In  experimental 
work  more  accurate  methods  of  measuring  the  velocity  are 
necessary,  and  for  this  purpose  the  boat  may  run  between 
buoys  whose  distance  apart  has  been  found  by  triangulation 
from  measured  bases  on  shore. 

When  a  boat  or  ship  is  to  be  propelled  through  water, 
the  resistances  to  be  overcome  increase  with  its  velocity, 
and  consequently,  as  in  railroad  trains,  a  practical  limit  of 
speed  is  soon  attained.  These  resistances  consist  of  three 
kinds — the  dynamic  pressure  caused  by  the  relative  velocity 
of  the  boat  and  the  water,  the  frictional  resistance  of  the 
surface  of  the  boat,  and  the  wave  resistance.  The  first  of 
these  can  be  entirely  overcome,  as  indicated  in  Art.  146,  by 
giving  to  the  boat  a  "fair  "  form,  that  is,  such  a  form  that 
the  dynamic  pressure  of  the  impulse  near  the  bow  is  bal- 
anced by  that  of  the  reaction  of  the  water  as  it  closes  in 
around  the  stern.  It  will  be  supposed  in  the  following  pages 
that  the  boat  has  this  form,  and  hence  this  first  resistance 
need  not  be  further  considered.  The  second  and  third 
sources  of  resistance  will  be  discussed  later. 

The  total  force  of  resistance  which  exists  when  a  vessel 
is  propelled  with  the  velocity  v  can  be  ascertained  by  draw- 
ing it  in  tow  at  the  same  velocity,  and  placing  on  the  tow 
line  a  dynamometer  to  register  the  tension.  An  experi- 
ment by  Froude  on  the  Greyhound,  a  steamer  of  1157  tons, 
gave  for  the  total  resistance  the  following  figures :  * 

*  Thearle's  Theoretical  Naval  Architecture  (London,  1876),  p.  347. 


476  NAVAL  HYDROMECHANICS  CHAP,  xv 

at  4  knots  per  hour,  0.6  tons ; 
at  6  knots  per  hour,  1.4  tons; 
at  8  knots  per  hour,  2 . 5  tons ; 
at  10  knots  per  hour,  4.7  tons; 
at  12  knots  per  hour,  9.0  tons. 

This  shows  that  at  low  speeds  the  resistance  varies  about  as 
the  square  of  the  velocity,  and  at  higher  speeds  in  a  faster 
ratio.  For  speeds  of  15  to  25  knots  per  hour,  the  usual  ve- 
locity of  ocean  steamers,  the  law  of  resistance  is  not  so  well 
known,  but  as  an  approximation  it  is  usually  taken  as  vary- 
ing with  the  square  of  the  velocity. 

Prob.  174.  What  horse-power  was  expended  in  the  above 
test  of  the  Greyhound  when  the  speed  was  12  knots  per  hour? 

ART.  175.     FRICTIONAL  RESISTANCES 

When  a  stream  or  jet  moves  over  a  surface  its  velocity 
is  retarded  by  the  frictional  resistances,  or  if  the  velocity 
be  maintained  uniform  a  constant  force  is  overcome.  In 
pipes,  conduits,  and  channels  of  uniform  section  the  velocity 
is  uniform,  and  consequently  each  square  foot  of  the  surface 
or  bed  exerts  a  constant  resisting  force,  the  intensity  of 
which  will  now  be  approximately  computed.  This  resist- 
ance will  be  the  same  as  the  force  required  to  move  the  same 
surface  in  still  water,  and  hence  the  results  will  be  directly 
applicable  to  the  propulsion  of  ships. 

Let  F  be  the  force  of  frictional  resistance  per  square  foot 
of  surface  of  the  bed  of  a  channel,  p  its  wetted  perimeter,  I 
its  length,  h  its  fall  in  that  length,  a  the  area  of  its  cross- 
section,  and  v  the  mean  velocity  of  flow.  The  force  of  fric- 
tion over  the  entire  surface  then  is  Fpl,  and  the  work  per 
second  lost  in  friction  is  Fplv.  The  work  done  by  the  water 
per  second  is  Wh  or  wavh.  Equating  these  two  expressions 
for  the  work,  there  results 

F  =w(a/p)  (h/l)  =  wrs 


ART.  175  FlUCTIOXAL    RESISTANCES  477 

in  which  r  is  the  hydraulic  radius  and  5  the  slope  of  the  water 
surface.  Now  inserting  for  rs  its  value  from  formula  (106), 
there  results 

F=wv2/c2 

in  which  w  is  the  weight  of  a  cubic  foot  of  water  and  c  is  the 
coefficient  in  the  Chezy  formula,  the  values  of  which  are 
given  in  Chapter  IX  and  the  accompanying  tables.  Inas- 
much as  the  velocities  along  the  bed  of  a  channel  are  some- 
what less  than  the  mean  velocity  v,  the  values  of  F  thus 
determined  will  probably  be  slightly  greater  than  the  actual 
resistance. 

For  smooth  iron  pipes  the  following  are  values  of  the 
frictional  resistance  in  pounds  per  square  foot  of  surface  at 
different  velocities,  as  computed  from  the  above  formula: 

Velocity,  feet  per  second          =2  4  6  10  15 

for  i  foot  diameter,  F=o.O23       0.080       0.17       0.43       0.92 

for  4  feet  diameter,  F=o.oi5       0.053       °-11       0.28       0.59 

These  figures  indicate  that  the  resistance  is  subject  to  much 
variation  in  pipes  of  different  diameters;  it  is  not  easy  to 
conclude  from  them,  or  from  formula  (106),  what  the  force 
of  resistance  is  for  plane  surfaces  over  which  water  is  moving. 

Experiments  made  by  moving  flat  plates  in  still  water  so 
that  the  direction  of  motion  coincides  with  the  plane  of  the 
surface  have  furnished  conclusions  regarding  the  laws  of 
fluid  friction  similar  to  those  deduced  from  the  flow  of  water 
in  pipes.  It  is  found  that  the  total  resistance  is  approxi- 
mately proportional  to  the  area  of  the  surface,  and  approxi- 
mately proportional  to  the  square  of  the  velocity.  Accord- 
ingly the  force  of  resistance  per  square  foot  may  be  written 

F=fv\  (175) 

in  which  v  is  the  velocity  in  feet  per  second  and  /  is  a  number 
depending  upon  the  nature  of  the  surface.  The  following 
are  average  values  of  /  for  large  surf  aces,  as  given  byllnwin  :* 

*  Encyclopaedia  Britannica,  Ninth  Edition,  vol.  12,  p.  483. 


478  NAVAL  HYDROMECHANICS  CHAP,  xv 

Varnished  surface,  /  =  o. 00250 

Painted  and  planed  plank,  /  =  0.00339 

Surface  of  iron  ships,  /  =  0.003 51 

Fine  sand  surface,  /  =  o. 00405 

New  well-painted  iron  plate,  /  =  0.0047 3 

Undoubtedly  the  value  of  /  is  subject  to  variations  with  the 
velocity,  but  the  experiments  on  record  are  so  few  that  the 
law  and  extent  of  its  variation  cannot  be  formulated.  It 
should,  however,  be  remarked  that  the  formulas  and  con- 
stants here  given  do  not  apply  to  low  velocities,  for  the  rea- 
sons given  in  Art.  116.  At  the  same  time  they  are  only  ap- 
proximately applicable  to  high  velocities.  A  low  velocity 
of  a  body  moving  in  an  unlimited  stream  may  be  regarded 
as  i  foot  per  second  or  less,  a  high  velocity  as  25  or  30  feet  per 
second. 

It  may  be  noted  that  the  above-mentioned  experiments 
indicate  that  the  value  of  F  is  greater  for  small  surfaces  than 
for  large  ones.  For  instance,  a  varnished  board  50  feet  long 
gave  /  =0.00250,  while  one  20  feet  long  gave  /  =0.00278,  and 
one  8  feet  long  gave  /  =  0.00325,  the  motion  being  in  all  cases 
in  the  direction  of  the  length.  The  resistance  is  the  same 
whatever  be  the  depth  of  immersion,  for  the  friction  is  unin- 
fluenced by  the  intensity  of  the  static  pressure.  This  is 
proved  by  the  circumstance  that  the  flow  of  water  in  a  pipe 
is  found  to  depend  only  upon  the  head  on  the  outlet  end, 
and  not  upon  the  pressure-heads  along  its  length. 

The  frictional  resistance  of  a  boat  or  ship  may  be  roughly 
estimated  by  taking  o.oo4^2  and  multiplying  it  by  the  im- 
mersed area.  For  instance,  if  this  area  be  8000  square  feet, 
the  frictional  resistance  at  a  velocity  of  10  feet  per  second 
is  3200  pounds,  but  at  a  velocity  of  20  feet  per  second  it  is 
1 2  800  pounds ;  the  horse -powers  needed  to  overcome  these 
resistances  are  58  and  464  respectively.  To  these  must  be 
added  the  power  necessary  to  overcome  the  friction  of  the 
air  and  that  wasted  in  the  production  of  waves. 


ART.  176  WORK   REQUIRED    FOR    PROPULSION  479 

The  above  discussion  refers  to  the  case  of  boats  moving 
in  the  ocean  and  lakes  or  in  a  stream  of  large  width  and 
depth.  In  a  canal  the  resistance  is  much  greater,  and  it 
depends  upon  the  ratio  of  the  cross-section  of  the  canal  to 
that  of  the  immersed  portion  of  the  boat.  When  the  width 
of  the  canal  is  about  five  times  that  of  the  boat  and  the  area 
of  its  cross-section  about  seven  times  that  of  the  boat,  the 
resistance  is  but  slightly  greater  than  in  an  unlimited  stream. 
For  smaller  ratios  the  resistance  rapidly  increases,  and  when 
two  boats  pass  each  other  in  a  small  canal  the  utmost  power 
of  the  horses  may  be  severely  taxed.  The  reason  for  this 
increased  resistance  appears  to  be  largely  due  to  the  fact 
that  the  velocity  of  the  water  relative  to  the  boat  increases 
with  the  diminution  of  the  cross-section  of  the  canal. 
Thus,  if  a  and  A  be  the  areas  of  the  cross-section  of  the 
canal  and  of  the  immersed  part  of  the  boat,  the  effective 
area  of  the  water  cross-section  is  a  —  A ,  and  the  water  flow- 
ing backward  through  this  area  must  have  a  higher  relative 
velocity  as  A  increases.  The  value  of  F  given  by  formula 
(175)  is  accordingly  increased  to  fv2/(i  —  (A/a))2. 

Prob.  175a.  What  horse-power  is  required  to  overcome  the 
frictional  resistance  of  a  boat  moving  at  the  rate  of  9  knots  per 
hour  when  the  area  of  its  immersed  surface  is  320  square  feet? 

Prob.  1756.  A  canal  has  a  cross-section  of  360  square  feet, 
while  that  of  a  canal  boat  is  60  square  feet.  Show  that  when 
two  boats  pass  each  other  the  resistance  of  each  is  increased 
about  60  percent. 

ART.  176.     WORK  REQUIRED  FOR  PROPULSION 

When  a  boat  or  ship  moves  through  still  water  with  a 
velocity  v,  it  must  overcome  the  pressure  due  to  impulse  of 
the  water  and  the  resistance  due  to  the  friction  of  its  surface 
on  the  water  and  air.  If  the  surface  be  properly  curved, 
there  is  no  resultant  pressure  due  to  impulse,  as  shown  in 
Art.  146.  The  resistance  caused  by  friction  of  the  im- 
mersed surface  on  the  water  can  be  estimated,  as  explained 


480  NAVAL  HYDROMECHANICS  CHAP,  xv 

above.  If  A  be  the  area  of  this  surface  in  square  feet,  the 
work  per  second  required  to  overcome  this  resistance  is 

k=AFv=fAv*  (176) 

The  work,  and  hence  the  horse-power,  required  to  move  a 
boat  accordingly  varies  approximately  as  the  cube  of  its 
velocity.  By  the  help  of  the  values  of  /  given  in  the  last 
article  an  approximate  estimate  of  the  work  can  be  made 
for  particular  cases.  The  resistance  of  the  air,  which  in 
practice  must  be  considered,  will  be  here  neglected. 

To  illustrate  this  law  let  it  be  required  to  find  how  many 
tons  of  coal  will  be  used  by  a  steamer  in  making  a  trip  of 
3000  miles  in  6  days,  when  it  is  known  that  800  tons  are 
used  in  making  the  trip  in  10  days.  As  the  power  used  is 
proportional  to  the  amount  of  coal,  and  as  the  distances 
traveled  per  day  in  the  two  cases  are  500  miles  and  300 
miles,  the  law  gives 

7/480  =  (5/3) 3 

whence  T  =  2220  tons.  By  the  increased  speed  the  expense 
for  fuel  is  increased  277  percent,  while  the  time  is  reduced  40 
percent.  If  the  value  of  wages,  maintenance,  interest,  etc., 
saved  on  account  of  the  reduction  in  time,  will  balance  the 
extra  expense  for  fuel,  the  increased  speed  is  profitable. 
That  such  a  compensation  occurs  in  many  instances  is  ap- 
parent from  the  constant  efforts  to  reduce  the  time  of  trips 
of  passenger  steamers. 

When  a  boat  moves  with  the  velocity  v  in  a  current 
which  has  a  velocity  u  in  the  same  direction  the  velocity  of 
the  boat  relative  to  the  water  is  v—u,  and  the  resistance  is 
proportional  to  (v  —  u)2  and  the  work  to  (v  —u) 3.  If  the  boat 
moves  in  the  opposite  direction  to  the  current  the  relative 
velocity  is  v  +  u,  and  of  course  v  must  be  greater  than  u  or 
no  progress  would  be  made.  In  all  cases  of  the  application 
of  the  formulas  of  this  article  and  the  last,  v  is  to  be  taken 
as  the  velocity  of  the  boat  relative  to  the  water. 


ART.  177  THE  JET  PROPELLER  481 

Another  source  of  resistance  to  the  motion  of  boats  and 
ships  is  the  production  of  waves.  This  is  due  in  part  to  a 
different  level  of  the  water  surface  along  the  sides  of  the  ship 
due  to  the  variation  in  static  pressure  caused  by  the  velocity, 
and  in  part  to  other  causes.  It  is  plain  that  waves,  eddies, 
and  foam  cause  energy  to  be  dissipated  in  heat,  and  that 
thus  a  portion  of  the  work  furnished  by  the  engines  of  the 
boat  is  lost.  This  source  of  loss  is  supposed  to  consume 
from  10  to  40  percent  of  the  total  work,  and  it  is  known  to 
increase  with  the  velocity.  On  account  of  the  uncertainty 
regarding  this  resistance,  as  well  as  those  due  to  the  friction 
of  the  water  and  air,  practical  computations  on  the  power 
required  to  move  boats  at  given  velocities  can  only  be  ex- 
pected to  furnish  approximate  results. 

The  investigations  of  Rankine  on  this  difficult  subject 
led  to  the  conclusion  announced  in  1858  in  the  anagram 
given  in  Prob.  i.  The  meaning  of  this  is  given  in  the  fol- 
lowing sentence,  published  in  1861:  "The  resistance  of  a 
sharp-ended  ship  exceeds  the  resistance  of  a  current  of  water 
of  the  same  velocity  in  a  channel  of  the  same  length  and 
mean  girth  by  a  quantity  proportional  to  the  square  of  the 
greatest  breadth  divided  by  the  square  of  the  length  of  the 
bow  and  stern." 

Prob.  176a.  How  many  tons  of  coal  are  required  to  make  a 
trip  in  4  days  if  650  tons  are  used  in  making  the  trip  in  5  days? 

Prob.  1766.  Compute  the  horse-power  required  to  maintain 
a  velocity  of  18  knots  per  hour,  taking  ^.  =  7473  square  feet 
and  /  =  0.004. 

ART.  177.     THE  JET  PROPELLER 

The  method  of  jet  propulsion  consists  in  allowing  water 
to  enter  the  boat  and  acquire  its  velocity,  and  then  to  eject 
it  backwards  at  the  stern  by  means  of  a  pump.  The  reac- 
tion thus  produced  propels  the  boat  forward.  To  investigate 
the  efficiency  of  this  method,  let  W  be  the  weight  of  water 


482  NAVAL  HYDROMECHANICS  CHAP,  xv 

ejected  per  second,  V  its  velocity  relative  to  the  boat,  and  v 
the  velocity  of  the  boat  itself.  The  absolute  velocity  of  the 
issuing  water  is  then  V  —  v,  and  it  is  plain  without  further 
discussion  that  the  maximum  efficiency  will  be  obtained 
when  this  is  o,  or  when  V  =  v,  as  then  there  will  be  no  energy 
remaining  in  the  water  which  is  propelled  backward.  It  is, 
however,  to  be  shown  that  this  condition  can  never  be  real- 
ized and  that  the  efficiency  of  jet  propulsion  is  low. 

The  effective  work  which  is  exerted  on  the  boat  by  the 
reaction  of  the  issuing  water  is 


g 

and  the  work  lost  in  the  absolute  velocity  of  the  water  is 


The  sum  of  these  is  the  total  theoretic  work,  or 

V2  -  v2 

Therefore  the  efficiency  of  jet  propulsion  is  expressed  by 

k  2V\ 

=K=V+^} 

This  becomes  equal  to  unity  when  v  =  V  as  before  indicated, 
but  then  it  is  seen  that  the  work  k  becomes  o  unless  W  is 
infinite.  The  value  of  W  is  waV,  if  a  be  the  area  of  the  ori- 
fices through  which  the  water  is  ejected;  and  hence  in  order 
to  make  e  unity  and  at  the  same  time  perform  work  it  is 
necessary  that  either  V  or  a  should  be  infinity.  The  jet 
propeller  is  therefore  like  a  reaction  wheel  (Art.  163),  and  it 
is  seen  upon  comparison  that  the  formula  for  efficiency  is 
the  same  in  the  two  cases. 

By  equating  the  above  value  of  the  useful  work  to  that 
established  in  the  last  article  there  is  found 

fgAv2=waV(V-v) 


ART.  178  PADDLE    WHEELS  483 

and  if  this  be  solved  for  V,  and  the  resulting  value  be  substi- 
tuted in  the  formula  for  e,  it  reduces  to 

c_  4 

3  +  Vi  +  (4fgA/wa) 

which  again  shows  that  e  approaches  unity  as  the  ratio  of  a 
to  A  increases.  The  area  of  the  orifices  of  discharge  must 
hence  be  very  large  in  order  to  realize  both  high  power  and 
high  efficiency.  For  this  reason  the  propulsion  of  vessels 
by  this  method  has  not  proved  economical,  although  in  the 
case  of  the  boat  Waterwitch,  built  in  England  about  1860, 
a  fair  speed  was  attained.  In  nature  the  same  result  is 
seen,  for  no  marine  animal  except  the  cuttle -fish  uses  this 
principle  of  propulsion.  Even  the  cuttle-fish  cannot  de- 
pend upon  his  jet  to  escape  from  his  enemies,  but  for  this 
relies  upon  his  supply  of  ink  with  which  he  darkens  the 
water  about  him. 

Prob.  177.  Compute  the  velocity  and  efficiency  of  a  jet  pro- 
peller driven  by  a  i-inch  nozzle  under  a  pressure  of  150  pounds 
per  square  inch  when  A  =  1000  square  feet  and  /  =  0.004.  Com- 
pute also  the  efficiency  when  the  diameter  of  the  nozzle  is  3 
inches. 

ART.  178.      PADDLE  WHEELS 

The  method  of  propulsion  by  rowing  and  paddling  is  well 
known  to  all.  The  power  is  furnished  by  muscular  energy 
within  the  boat,  the  water  is  the  fulcrum  upon  which  the 
blade  of  the  oar  acts,  and  the  force  of  reaction  thus  produced 
is  transmitted  to  the  boat  and  urges  it  forward.  If  water 
were  an  unyielding  substance,  the  theoretic  efficiency  of  the 
oar  should  be  unity,  or,  as  in  any  lever,  the  work  done  by 
the  force  at  the  rowlock  should  equal  the  work  performed 
by  the  motive  force  exerted  by  the  man  on  the  handle  of 
the  oar.  But  as  the  water  is  yielding,  some  of  it  is  driven 
backward  by  the  blade  of  the  oar,  and  thus  energy  is  lost. 

The  paddle  or  side  wheel  so  extensively  used  in  river 
navigation  is  similar  in  principle  to  the  oar.  The  power 


484  NAVAL  HYDROMECHANICS  CHAP,  xv 

is  furnished  by  a  motor  within  the  boat,  the  blades  or  vanes 
of  the  wheel  tend  to  drive  the  water  backward,  and  the  reac- 
tion thus  produced  urges  the  boat  forward.  On  first  thought 
it  might  be  supposed  that  the  efficiency  of  the  method  would 
be  governed  by  laws  similar  to  those  of  the  undershot  wheel, 
and  such  would  be  the  case  if  the  vessel  were  stationary  and 
the  wheel  were  used  as  an  apparatus  for  moving  the  water. 
In  fact,  however,  the  theoretic  efficiency  of  the  paddle  wheel 
is  much  higher  than  that  of  the  undershot  motor. 

The  work  exerted  by  the  steam-engine  upon  the  paddle 
wheels  may  be  represented  by  PV,  in  which  P  is  the  pressure 
produced  by  the  vanes  upon  the  water,  and  V  is  their  ve- 
locity of  revolution  ;  and  the  work  actually  imparted  to  the 
boat  may  be  represented  by  Pv,  in  which  v  is  its  velocity 
with  respect  to  the  water.  Accordingly  the  efficiency  of 
the  paddle  wheel,  neglecting  losses  due  to  foam  and  waves,  is 


- 

V 

in  which  vl  is  the  difference  V  —  v,  or  the  so-called  "  slip."  If 
the  slip  be  o,  the  velocities  V  and  v  are  equal,  and  the  theo- 
retic efficiency  is  unity.  The  value  of  V  is  determined  from 
the  radius  r  of  the  wheel  and  its  number  of  revolutions  per 
second;  thus  V  =  2nrn. 

On  account  of  the  lack  of  experimental  data  it  is  difficult 
to  give  information  regarding  the  practical  efficiency  of  pad- 
dle wheels  considered  from  a  hydromechanic  point  of  view. 
Owing  to  the  water  which  is  lifted  by  the  blades,  and  to  the 
foam  and  waves  produced,  much  energy  is  lost.  They  are, 
however,  very  advantageous  on  account  of  the  readiness 
with  which  the  boat  can  be  stopped  and  reversed.  When 
the  wheels  are  driven  by  separate  engines,  as  is  sometimes 
done  on  river  boats,  perfect  control  is  secured,  as  they  can 
be  revolved  in  opposite  directions  when  desired.  Paddle 
wheels  with  feathering  blades  are  more  efficient  than  those 
with  fixed  radial  ones,  but  practically  they  are  found  to  be 


ART.  179  THE   SCREW   PROPELLER  485 

cumbersome,  and  liable  to  get  out  of  order.  In  ocean  navi- 
gation the  screw  has  now  almost  entirely  replaced  the  paddle 
wheel  on  account  of  its  higher  efficiency. 

Prob.  178.  The  radius  of  the  blades  of  a  paddle  wheel  is 
10.5  feet  and  the  number  of  revolutions  per  minute  is  24.  If 
the  efficiency  is  75  percent,  what  is  the  velocity  of  the  boat  in 
miles  per  hour  ?  Show  that  for  this  case  the  slip  is  33  percent 
of  the  velocity  of  the  boat. 

ART.  179.     THE  SCREW  PROPELLER 

The  screw  propeller  consists  of  several  helicoidal  blades 
attached  at  the  stern  of  a  vessel  to  the  end  of  a  horizontal 
shaft  which  is  made  to  revolve  by  steam  power.  The  dy- 
namic pressure  of  the  reaction  developed  between  the  water 
and  the  helicoidal  surface  drives  the  vessel  forward,  the  theo- 
retic work  of  the  screw  being  the  product  of  this  pressure 
by  the  distance  traversed.  The  pitch  of  the  screw  is  the 
distance,  parallel  to  the  shaft,  between  any  point  on  a  helix, 
and  the  corresponding  point  on  the  same  helix  after  one  turn 
around  the  axis,  and  the  pitch  may  be  constant  at  all  dis- 
tances from  the  axis,  or  it  may  be  variable.  If  the  water 
were  unyielding,  the  vessel  would  advance  a  distance  equal 
to  the  pitch  at  each  revolution  of  the  shaft;  actually,  the 
advance  is  less  than  the  pitch,  the  difference  being  called  the 
"  slip."  The  effect  thus  is  that  the  pressure  P  existing  be- 
tween the  helical  surfaces  and  the  water  moves  the  vessel 
with  the  velocity  v,  while  the  theoretic  velocity  which  should 
occur  is  V,  being  the  pitch  of  the  screw  multiplied  by  the 
number  of  revolutions  per  second.  The  work  expended  is 
hence  PV  or  P(v  +  v^,  if  vl  be  the  "  slip"  per  second,  and  the 
work  utilized  is  Pv.  Accordingly  the  efficiency  of  screw 
propulsion  is,  approximately, 


which  is  the  same  expression  as  before  found  for  the  paddle 
wheel.     Here,  as  in  the  last  article,  all  the  pressure  exerted 


486  NAVAL  HYDROMECHANICS  CHAP,  xv 

by  the  blades  upon  the  water  is  supposed  to  act  backward 
in  a  direction  parallel  to  the  shaft  of  the  screw,  and  the  above 
conclusion  is  approximate  because  this  is  actually  not  the 
case,  and  also  because  the  action  of  friction  has  not  been 
considered. 

The  pressure  P  which  is  exerted  by  the  helicoidal  blades 
upon  the  water  is  the  same  as  the  thrust  or  stress  in  the  shaft, 
and  the  value  of  this  may  be  approximately  ascertained  by 
regarding  it  as  due  to  the  reaction  of  a  stream  of  water  of 
cross-section  a  and  velocity  v,  or 


Another  expression  for  this  may  be  found  from  the  indicated 
work  k  of  the  steam  cylinders  of  the  engines  ;  thus 

P=k/v 

Numerical  values  computed  from  these  two  expressions  do 
not,  however,  agree  well,  the  latter  giving  in  general  a  much 
less  value  than  the  former. 

In  Art.  176  the  work  to  be  performed  in  propelling  a 
vessel  of  fair  form  having  the  submerged  surface  A  was 

found  to  be 

k=fAv3 

If  the  value  of  v  is  taken  from  this  and  inserted  in  the  ex- 
pression for  efficiency,  there  obtains 


which  shows  that  e  increases  as  vlt  /,  and  A  decrease,  and 
as  k  increases.  Or  for  given  values  of  /  and  A  the  efficiency 
decreases  with  the  speed. 

It  has  been  observed  in  a  few  instances  that  the  "  slip" 
vl  is  negative,  or  that  V,  as  computed  from  the  number  of 
revolutions  and  pitch  of  the  screw,  is  less  than  v.  This  is 
probably  due  to  the  circumstance  that  the  water  around 
the  stern  is  following  the  vessel  with  a  velocity  v'  ',  so  that 
the  real  slip  is  V—  v  +  vf  instead  of  V—v.  The  existence 


ART.  180  STABILITY  OF  A  SHIP  487 

of  negative  slip  is  usually  regarded  as  evidence  of  poor 
design. 

In  some  cases  twin  screws  are  used,  as  with  these  the 
vessel  can  be  more  readily  controlled.  Fig.  179  shows  the 
twin  screws  of  the  New  York,  an  ocean  steamer  of  580  feet 


FIG.  179 

length,  63.5  feet  breadth,  and  42  feet  depth,  with  a  gross 
tonnage  of  10  500  and  an  estimated  horse-power  of  about 
1 6  ooo.  These  screws  revolve  in  opposite  directions.  The 
practical  advantage  of  the  screw  over  the  paddle  wheel  has 
been  found  to  be  very  great,  and  this  is  probably  due  to  the 
circumstance  that  less  energy  is  wasted  in  lifting  the  water 
and  in  forming  waves. 

Prob.  179.  A  steamer  having  a  submerged  surface  of  30  ooo 
square  feet  is  propelled  at  18  knots  per  hour  by  an  expenditure 
of  6000  horse-powers.  If  the  pitch  of  the  screw  is  20  feet,  its 
number  of  revolutions  120  per  minute,  and  /  =  0.004,  compute 
the  number  of  lost  horse-powers. 

ART.  180.     STABILITY  OF  A  SHIP 

In  Art.  14  the  general  principles  regarding  the  stability 
of  a  floating  body  were  stated,  and  these  are  of  great  im- 
portance in  the  design  of  ships.  The  center  of  gravity  is, 
of  course,  always  above  the  center  of  buoyancy,  and  the 
metacenter  must  be  above  the  center  of  gravity  in  order  to 
ensure  stability.  The  distance  between  the  metacenter  and 
the  center  of  gravity  is  denoted  by  m,  and  if  the  body  be  in- 


488 


NAVAL  HYDROMECHANICS 


CHAP.  XV 


clined  slightly  to  the  vertical  at  the  angle  6,  the  moment  of 
the  couple  formed  by  the  weight  W  of  the  body  which  acts 
downward  through  the  center  of  gravity  and  the  upward 
pressure  W  of  the  displaced  water  which  acts  through  the 
center  of  buoyancy  is  Wm  tan  6.  Hence  m  tan  6  is  a  measure 
of  the  stability  of  the  body,  and  the  greater  its  value  the 
greater  is  the  tendency  of  the  body  to  return  to  the  upright 
position. 

The  metacentric  height  m  cannot,  however,  be  made  very 
great,  for  the  rapidity  of  rolling  increases  with  it.  When  a 
floating  body  or  ship  is  displaced  from  its  vertical  position 
it  rolls  to  and  fro  with  isochronous  oscillations  like  those 
of  a  pendulum  and  the  time  of  one  oscillation  from  port 
to  starboard  is  given  by  the  formula 

t  =  xVr2/mg 

in  which  r  is  the  radius  of  gyration  of  the  weight  of  the  ship 
about  a  horizontal  axis  passing  through  its  center  of  gravity. 
Hence  if  m  be  large,  t  is  small  and  the  ship  rolls  quickly, 
but  if  m  be  small,  t  is  large  and  the  ship  rolls  slowly.  The 
metacentric  height  m  for  ocean  vessels  usually  ranges  from 
2  to  15  feet,  about  6  or  8  feet  being  the  usual  value. 


FIG.  180<z 


FIG.  1806 


.  The  determination  of  the  values  of  m  and  r  for  a  ship  is 
a  laborious  process  owing  to  its  curved  shape  and  the  irregu- 
lar distribution  of  its  weight  and  cargo.  The  process  will 
here  be  applied  to  the  simple  case  of  a  rectangular  prism  of 
uniform  density.  Let  h  be  the  height  and  b  the  breadth 
of  the  prism,  and  I  its  length  perpendicular  to  the  plane 


ART.  180  STABILITY  OF  A  SHIP  489 

of  the  drawing  in  Fig.  180a.  When  the  prism  is  in  the  ver- 
tical position  its'  depth  of  flotation  is  sh,  Us  be  its  specific 
gravity  (Art.  13),  and  this  is  also  the  length  of  the  immersed 
portion  of  the  axis  A  B  when  the  prism  is  inclined  to  the 
vertical  at  the  angle  6,  as  in  Fig.  1806.  In  the  latter  posi- 
tion the  center  of  buoyancy  D,  being  the  center  of  gravity 
of  the  displaced  water,  is  easily  located,  and 
_62tan0  _sh  b2  tan2/? 

I2Sk  ~    2  24Sk 

are  its  coordinates  with  respect  to  B,  %  being  measured  nor- 
mal and  y  parallel  to  AB.  The  distance  m  from  the  center 
of  gravity  g  to  the  metacenter  M  is  then  found  to  be 


If  m  is  positive  the  metacenter  is  above  the  center  of  gravity 
and  the  equilibrium  is  stable,  for  the  moment  Wm  tan0  re- 
stores the  prism  to  the  vertical  position;  if  m  is  zero  the 
equilibrium  is  indifferent;  if  m  is  negative  the  equilibrium 
is  unstable  and  the  prism  falls  over. 

The  square  of  the  radius  of  gyration  of  the  prism  with 
respect  to  a  horizontal  longitudinal  axis  through  G  is  its 
polar  moment  of  inertia  ^I(bh3  +  hb3)  divided  by  its  volume 
Ibd,  whence  r2  =  ^(h2  +  b2).  For  example,  if  h  be  5  feet,  b  be 
8  feet,  and  5  be  0.5,  the  value  of  r2  is  7:42  feet2.  The  value 
of  m  to  be  used  in  the  above  formula  for  the  time  of  one 
roll  is  that  obtained  by  making  0  equal  to  zero,  since  that 
formula  is  strictly  true  only  for  small  deviations  from  the 
vertical.  For  the  above  data  this  value  of  m  is  +0.88  feet, 
the  plus  sign  denoting  stability,  and  hence  the  time  of  one 
oscillation  from  port  to  starboard  is  t  =  i.6i  seconds.  It  is 
seen  that  t  can  be  increased  either  by  increasing  r2  or  by  de- 
creasing m  ;  since  a  decrease  in  m  is  unfavorable  to  stability 
it  is  often  preferable  to  increase  r2.  For  instance,  in  loading 
a  ship  the  cargo  may  be  placed  along  the  sides  rather  than 
near  the  middle  of  the  hold,  and  this  will  increase  r2,  as  the 


490  NAVAL  HYDROMECHANICS  CHAP,  xv 

width  of  a  ship  is  always  greater  than  its  depth.  The  gen- 
eral rule  to  promote  stability  and  prevent  quick  rolling  is 
hence  to  place  the  cargo  as  far  as  possible  from  the  center 
of  gravity. 

The  above  formula  for  m  shows  that  the  moment  Wm 
tan/9  which  restores  the  floating  prism  to  the  vertical  in- 
creases with  the  angle  6  up  to  a  maximum  value,  then  de- 
creases, and  when  D  arrives  vertically  beneath  G  it  becomes 
zero  and  the  prism  upsets.  For  the  case  where  h  =  5  feet, 
6=8  feet,  and  s  =  0.5,  the  value  of  m  tan#  is  o.oo  feet  for 
0  =  0°,  o.i 6  feet  for  6  =  10°,  0.37  feet  for  6  =  20°,  and  0.72 
feet  for  6  =  30°;  at  6  =32°  the  corner  of  the  prism  becomes 
immersed  so  that  the  formula  no  longer  holds,  but  up  to  this 
point  the  moment  constantly  increases.  From  the  above 
expression  for  m  the  solution  of  the  two  cases  of  Prob.  14 
is  readily  made,  but  the  condition  given  for  the  second 
case  holds  good  only  when  no  part  of  the  top  of  the  prism 
is  immersed. 

Prob.  180a.  Deduce  the  above  values  of  x,  y,  and  m. 

Prob.  1806.  An  open  rectangular  wooden  box  caisson  of 
length  /,  breadth  6,  and  depth  d  has  sides  of  mean  thickness  bl 
and  a  bottom  of  thickness  d^.  Deduce  formulas  for  the  meta- 
centric  height  m  and  the  squared  radius  of  gyration  r2.  Com- 
pute m,  r2,  and  t  for  a  numerical  case. 

ART.  181.      ACTION  OF  THE  RUDDER 

The  action  of  the  rudder  in  steering  a  vessel  involves  a 
principle  that  deserves  discussion.  In  Fig.  181  is  shown  a 

plan  of  a  boat  with  the  rudder 
turned  to  the  starboard  side,  at  an 
angle  d  with  the  line  of  the  keel. 
The  velocity  of  the  vessel  being  v, 
the  action  of  the  water  upon  the 
rudder  is  the  same  as  if  the  vessel 
FIG.  181  were  at  rest  and  the  water  in 

motion  with  the  velocity  v.     Let  W  be  the  weight  of  water 


ART.  181  ACTION    OF   THE   RUDDER  491 

which  produces  dynamic  pressure  against  the  rudder,  due 
to  the  impulse  W.  v/g  (Art.  143).  The  component  of  this 
pressure  normal  to  the  rudder  is 

P  =  Wv  sind/g 

and  its  effect  in  turning  the  vessel  about  the  center  of  gravity 
C  is  measured  by  its  moment  with  reference  to  that  point. 
Let  b  be  the  breadth  of  the  rudder  and  d  the  distance  CH  be- 
tween the  center  of  gravity  and  the  hinge  of  the  rudder,  then 
the  lever  arm  of  the  force  P  is 


and  accordingly  the  turning  moment  is 


To  determine  that  value  of  6  which  produces  the  greatest 
effect  in  turning  the  boat  the  derivative  of  M  with  respect  to 
6  must  vanish,  which  gives 

b         II       W 

cos0  =  —  -&-7+\-       ~ 
Sd      *  2 


and  from  this  the  value  of  6  is  found  to  be  approximately 
45°,  since  d  is  always  much  larger  than  b. 

Values  of  the  angle  6  for  several  values  of  the  ratio  b/d 
may  now  be  computed  as  follows  : 

b/d=   i       I      &     Tfr      O 

cos0  =0.6825   0.6916   0.6947   0.7069   0.7071 
6  =46°  58'   46°  15'   46°  oo'   45°  01'    45° 

which  show  that  about  45°  is  the  advantageous  angle. 
In  practice  it  is  usual  to  arrange  the  mechanism  of  the 
rudder  so  that  it  can  only  be  turned  to  an  angle  of  about  42° 
with  the  keel,  for  it  is  found  that  the  power  required  to  turn 
it  the  additional  3°  or  4°  is  not  sufficiently  compensated 
by  the  slightly  greater  moment  that  would  be  produced. 
The  reasoning  also  shows  that  intensity  of  the  turning  mo- 
ment increases  with  v,  so  that  the  rudder  acts  most  promptly 
when  the  boat  is  moving  rapidly.  For  the  same  reason  a 


492  NAVAL  HYDROMECHANICS  CHAP,  xv 

rudder  on  a  steamer  propelled  by  a  screw  is  not  required  to 
be  so  broad  as  one  on  a  boat  driven  by  paddle  wheels,  for  the 
effect  of  the  screw  is  to  increase  the  velocity  of  the  im- 
pinging water,  and  hence  also  to  increase  the  dynamic  pres- 
sure against  the  rudder. 

Prob.  181.  Explain  how  it  is  that  a  boat  can  sail  against  the 
wind.     What  is  the  influence  of  the  keel  in  this  motion? 


ART.  182.     TIDES  AND  WAVES 

The  complete  discussion  of  the  subject  of  waves  might, 
like  so  many  other  branches  of  hydraulics,  be  expanded 
so  as  to  embrace  an  entire  treatise,  and  hence  there  can  be 
here  given  only  the  briefest  outline  of  a  few  of  the  most  im- 
portant principles.  There  are  two  classes  or  kinds  of  waves, 
the  first  including  the  tidal  waves  and  those  produced  by 
earthquakes  or  other  sudden  disturbances,  and  the  second 
those  due  to  the  wind.  The  daily  tidal  wave  generated  by 
the  attraction  of  the  moon  and  sun  originates  in  the  South 
Pacific  Ocean,  whence  it  travels  in  all  directions  with  a  ve- 
locity dependent  upon  the  depth  of  water  and  the  configu- 
ration of  the  continents,  and  which  in  some  regions  is  as  high 
as  1000  miles  per  hour.  Striking  against  the  coasts,  the 
tidal  waves  cause  currents  in  inlets  and  harbors,  and  if  the 
circumstances  were  such  that  their  motion  could  become 
uniform  and  permanent,  these  might  be  governed  by  the 
same  laws  which  apply  to  the  flow  of  water  in  channels. 
Such,  however,  is  rarely  the  case ;  and  accordingly  the  sub- 
ject of  tidal  currents  is  one  of  much  complexity  and  not 
capable  of  general  formulation. 

The  velocity  of  a  tidal  wave  on  the  ocean  is  \/gD  where 
D  is  the  depth  of  the  water.  When  such  a  wave  rolls  over 
the  land  the  greatest  velocity  it  can  have  is  Vgd,  where  d 
is  its  depth,  this  being  the  case  of  the  bore  (Art.  133).  The 
velocity  of  a  wave  produced  by  a  sudden  disturbance  in  a 


ART.  182  TlDES    AND    WAVES  493 


channel  of  uniform  width  has  also  been  found  to 
where  D  is  the  depth  of  the  water. 

Rolling  waves  produced  by  the  wind  travel  with  a  velocity 
which  is  small  compared  with  those  above  noted,  although 
in  water  where  the  disturbance  can  extend  to  the  bottom  it 
is  generally  supposed  that  their  velocity  is  VgD.  Upon 


FIG.  182 

the  ocean  the  maximum  length  of  such  waves  is  estimated 
at  550  feet  and  their  velocity  at  about  53  feet  per  second. 
For  this  class  of  waves  it  is  found  by  observation  that  each 
particle  of  water  upon  the  surface  moves  in  an  elliptic  or 
circular  orbit,  whose  time  of  revolution  is  the  same  as  the 
time  of  one  wave  length.  Thus  the  particles  on  the  crest 
of  a  wave  are  moving  forward  in  the  direction  of  the  mo- 
tion of  the  wave  while  those  in  the  trough  are  moving  back- 
ward. When  such  waves  advance  into  shallow  water  their 
length  and  speed  decrease,  but  the  time  of  revolution  of  the 
particles  in  their  orbits  remains  unaltered,  and  as  a  conse- 
quence the  slopes  become  steeper  and  the  height  greater, 
until  finally  the  front  slope  becomes  vertical  arid  the  wave 
breaks  with  roar  and  foam.  Below  the  surface  the  particles 
revolve  also  in  elliptic  orbits,  which  grow  smaller  in  size 
toward  the  bottom.  The  curve  formed  by  the  vertical  sec- 
tion of  the  surface  of  a  wave  at  right  angles  to  its  length  is 
of  a  cycloidal  nature. 

The  force  exerted  by  ocean  waves  when  breaking  against 
sea  walls  is  very  great,  as  already  mentioned  in  Art.  146,  and 
often  proves  destructive.  If  walls  can  be  built  so  that  the 
waves  are  reflected  without  breaking,  as  is  sometimes  possible 
in  deep  water,  their  action  is  rendered  less  injurious.  Upon 
the  ocean  waves  move  in  the  same  direction  as  the  wind,  but 


494  NAVAL  HYDROMECHANICS  CHAP,  xv 

along  shore  it  is  observed  that  they  move  normally  toward 
it,  whatever  may  be  the  direction  in  which  the  wind  is  blow- 
ing. The  force  of  wave  action  is  felt  at  depths  of  over  100 
feet  below  the  surface,  for  sand  has  been  brought  up  from 
depths  of  80  feet  and  dropped  upon  the  decks  of  vessels. 
Shoals  also  cause  a  marked  increase  in  the  height  of  waves 
even  when  such  shoals  are  500  feet  or  more  below  the 
water  surface. 

Prob.  182a.  In  a  channel  6.5  feet  wide,  and  of  a  depth  de- 
creasing 1.5  feet  per  1000  feet,  Bazin  generated  a  wave  by  sud- 
denly admitting  water  at  the  upper  end.  At  points  where  the 
depths  were  2.16,  1.85,  1.46,  and  0.80  feet,  the  velocities  were 
observed  to  be  8.70,  8.67,  7.80,  and  6.69  feet  per  second.  Do 
these  velocities  agree  with  the  theoretic  law? 

Prob.  1826.  Show  that  the  values  of  /  given  in  Art.  175  for 
use  in  the  formula  F  =  fv2  are  to  be  multiplied  by  5.255  when  v 
is  in  meters  per  second  and  F  in  kilograms  per  square  meter. 

Prob.  182c.  Compute  the  metric  horse-power  required  for 
a  velocity  of  25  kilometers  per  hour  for  a  boat  which  has  a 
submerged  area  of  237  square  meters. 

Prob.  lS2d.  A  ship  rolls  from  starboard  to  port  in  7.5  seconds. 
If  the  metacentric  height  m  is  2.4  meters,  what  is  the  value  of 
the  transverse  radius  of  gyration  of  the  ship  ?  How  much  must 
the  radius  of  gyration  be  increased  in  order  to  increase  the  time 
of  rolling  15  percent? 


ART.  183  GENERAL   NOTES   AND    PRINCIPLES  495 


CHAPTER  XVI 
PUMPS  AND  PUMPING 

ART.  183.     GENERAL  NOTES  AND  PRINCIPLES 

Among  the  simple  devices  for  raising  water  that  have 
been  used  for  many  centuries,  and  which  may  be  called  lift 
pumps  in  a  general  way,  are  the  sweep  and  windlass,  buckets 
attached  to  a  revolving  wheel,  the  chain  and  bucket  pump 
where  the  buckets  move  in  a  cylinder,  and  the  Archimedian 
screw.  The  chain  and  bucket  pump  was  probably  first  used 
by  the  Chinese  in  the  form  of  an  inclined  trough  in  which 
moved  the  buckets  attached  to  the  endless  chain,  and  this 
device  in  various  forms  has  been  used  in  all  countries  for 
lifting  water  from  wells.  The  Archimedian  screw,  invented 
by  the  great  engineer  Archimedes  when  he  was  in  Egypt, 
about  240  B.C.,  consists  of  a  tube  wound  spirally  around  an 
inclined  cylinder.  If  the  lower  end  be  placed  under  water 
and  the  cylinder  be  revolved  the  water  is  lifted  and  flows  out 
of  the  upper  end  of  the  tube.  This  screw  pump  is  still  in 
use  in  Egypt  and  it  is  said  to  be  an  efficient  apparatus  for  a 
low  lift. 

The  fact  that  water  would  sometimes  rise  into  a  space 
from  which  the  air  had  been  removed  was  known  at  a  re- 
mote antiquity,  and  this  was  frequently  explained  by  the 
statement  that  "  nature  abhors  a  vacuum."  It  was  not 
until  the  middle  of  the  seventeenth  century  that  the  true 
reason  of  this  phenomenon  was  explained  through  the 
researches  of  Torricelli  and  Pascal  (Art.  5),  but  prior  to  this 
time  a  rude  form  of  suction  pump,  made  by  attaching  a 
pipe  to  a  bellows  at  the  opening  where  the  air  usually  enters, 


496  PUMPS  AND  PUMPING  CHAP,  xvi 

was  used  in  both  France  and  Germany.  In  1732  the  first 
true  suction  and  lift  pump  was  devised  by  Boulogne,  and  a 
little  later  the  suction  and  force  pump  came  into  use. 

The  force  pump  is  a  device  for  raising  water  by  means 
of  pressure  exerted  on  it  by  a  piston.  The  syringe,  which 
has  been  known  from  very  early  times,  is  an  example  of 
this  principle,  but  the  first  true  force  pump  was  invented 
in  Egypt  about  250  B.C.,  by  Ctesibius,  a  Greek  hydraulician, 
and  the  description  of  'it  given  by  Vitruvius  indicates  that  it 
was  used  to  some  extent  by  the  Romans.  The  early  force 
pumps  were  placed  with  their  cylinders  below  the  level  of 
the  water  to  be  lifted  and  had  valves  which  closed  under  the 
back  pressure  of  the  water.  By  placing  the  cylinders  above 
the  water  level  and  utilizing  the  principle  of  suction  the 
suction  and  force  pump  originated. 

All  devices  for  raising  water  may  be  classified  under  the 
three  principles  above  mentioned,  that  of  lifting  in  buckets, 
drawing  it  up  by  suction,  or  forcing  it  up  by  pressure,  or 
under  combinations  of  these.  The  lift  or  bucket  principle 
is  mainly  employed  for  small  quantities  of  water  and  has 
only  a  limited  use  in  engineering  practice.  The  suction 
principle,  combined  with  lift  or  pressure,  is,  extensively  used, 
but  in  no  event  can  the  height  of  the  suction  exceed  34  feet, 
for  it  is  the  atmospheric  pressure  that  causes  the  water  to 
rise  when  the  air  above  it  is  exhausted ;  under  this  principle 
also  may  be  put  injector  pumps  which  operate  under  the 
action  of  negative  pressure-head  (Art.  32).  The  principle  of 
direct  pressure  governs  not  only  the  force  pump,  but  rotary 
and  centrifugal  pumps  and  also  the  devices  for  raising  water 
by  compressed  air. 

Whenever  water  is  raised  from  a  lower  to  a  higher  level 
an  amount  of  work  must  be  expended  greater  than  the 
theoretic  work  required  to  lift  the  given  weight  of  water 
through  the  given  height.  The  excess,  called  the  lost  work, 
is  spent  in  overcoming  resistances  of  friction  and  inertia. 
In  designing  pumps  it  is  the  object  to  reduce  these  losses 


ART.  183  GENERAL   NOTES   AND    PRINCIPLES  497 

to  a  minimum,  so  that  the  greatest  economy  in  operation 
may  result.  The  subject  will  here  be  mainly  considered 
from  a  hydraulic  standpoint,  the  object  being  to  set  forth  the 
fundamental  principles  by  which  hydraulic  losses  may  be 
avoided  as  far  as  possible. 

Let  W  be  the  weight  of  water  raised  per  second  and  h 
the  height  of  the  lift,  then  the  useful  work  per  second  k  is 
Wh.  Let  the  total  work  expended  per  second  be  called  K, 
then  the  efficiency  of  the  apparatus  is  e  =  k/K.  The  work 
K  to  be  considered  here  is  that  delivered  to  the  pump  and 
does  not  include  that  lost  in  transmission  from  the  motor, 
since  this,  of  course,  is  not  fairly  chargeable  against  the 
pump  or  lifting  apparatus.  If  K  be  replaced  by  W(h  +  hf) 
where  hf  is  the  head  lost  in  overcoming  the  frictional  resist- 
ances, then  the  efficiency  may  be  written 

k         h 

p  __  —  _  /I  OQ\ 

~K  ~        ' 


which  is  less  than  unity,  since  h'  cannot  be  made  zero. 

The  power  required  to  operate  a  pump  to  raise  the  weight 
W  of  water  per  second  through  the  height  h  is  easily  com- 
puted if  the  efficiency  of  the  pump  is  known.  For  ex- 
ample, to  raise  150  gallons  per  second  through  a  height  of 
20  feet  with  a  pump  having  an  efficiency  of  62  percent,  the 
work  imparted  to  the  pump  per  second  is 

K  =k/e  =  (i$o  X8.355  X2o)/o.62  =25  ooo  foot-pounds 
and  this,  divided  by  550,  gives  45.5  horse-powers. 

Prob.  183a.  What  is  the  efficiency  of  a  bucket  pump  which 
raises  840  gallons  of  water  per  minute  through  a  height  of  18 
feet  with  an  expenditure  of  5  horse-powers? 

Prob.  1836.  A  pump  raises  20.5  cubic  feet  of  water  per  second 
through  a  height  of  127.5  feet.  The  lost  head  in  the  pump  and 
pipes  amounts  to  13.5  feet.  Compute  the  efficiency  of  the  pump- 
ing plant  and  the  power  required  to  operate  it. 


498  PUMPS  AND  PUMPING  CHAP,  xvr 

ART.  184.     RAISING  WATER  BY  SUCTION 

The  term  suction  is  a  misleading  one  unless  it  be  clearly 
kept  in  mind  that  water  will  not  rise  in  a  vacuum  tube  unless 
the  atmospheric  pressure  can  act  underneath  it.  For  ex- 
ample, no  amount  of  rarefaction  above  the  surface  of  the 
water  in  a  glass  bottle  will  cause  that  water  to  rise.  When 
the  tube  is  inserted  into  a  river  or  pond,  however,  the  water 
will  rise  in  it  when  a  partial  vacuum  is  formed,  since  the 
atmospheric  pressure  which  is  transmitted  through  the 
water  pushes  it  up  until  equilibrium  is  secured  (Art.  5). 
The  mean  atmospheric  pressure  of  14.7  pounds  per  square 
inch  at  the  sea  level  is  equivalent  to  a  height  of  water  of 
34  feet,  and  this  is  the  limit  of  raising  water  by  suction 
alone.  In  practice  this  height  cannot  be  reached  on  account 
of  the  impossibility  of  producing  a  perfect  vacuum,  and  it  is 
found  that  about  28  feet  is  the  maximum  practicable  height, 
of  suction  lift. 

The  height  of  the  water  barometer  varies  with  the 
state  of  the  weather,  with  the  elevation  above  sea  level,, 
and  with  the  temperature.  The  value  of  34  feet  is  that 
corresponding  to  a  reading  of  30  inches  on  the  mercury 
barometer  at  a  temperature  of  32°  Fahrenheit.  For  higher 
temperatures  more  or  less  vapor  is  evaporated  from  the 
water  surface  and  fills  the  suction  tube,  so  that  a  complete 
vacuum  cannot  be  formed.  When  the  mercury  barometer 
reads  30  inches,  the  water  barometer  is  only  33.4  feet  if 
the  temperature  of  the  water  is  60°  Fahrenheit,  32.4  feet 
at  90°,  about  30  feet  for  120°,  about  23  feet  for  160°,  about. 
6  feet  for  200°,  and  for  212°  it  is  zero,  since  the  tube  is  filled 
with  steam.  Hence  water  at  the  boiling-point  cannot  be 
raised  by  suction. 

Fig.  184  gives  two  diagrams  illustrating  the  principle 
of  action  of  the  common  suction  and  lift  pump.  It  consists 
of  two  vertical  tubes  BD  and  BE,  the  former  being  called 
the  suction  pipe  and  the  latter  the  pump  cylinder.  The 


ART.  184 


RAISING  WATER  BY  SUCTION 


499 


piston  A  in  the  pump  cylinder  has  a  valve  opening  upward, 
and  the  valve  B  at  the  top  of  the  suction  pipe  also  opens 
upward.  In  the  left-hand 
diagram  the  piston  is  de- 
scending, the  valve  A  be- 
ing open  and  B  being 
closed  under  the  pressure 
of  the  air  in  the  space  be- 
tween them.  In  the  right- 
hand  diagram  the  piston 
is  ascending,  the  valve  A 
being  closed  by  the  pres- 
sure of  the  air  or  water 
above  it,  while  B  is  open 
owing  to  the  excess  of  at- 
mospheric pressure  in  BD 
above  that  in  AB.  In  the  FlG- 184 

first  diagram  the  piston  has  made  only  one  or  two  strokes 
so  that  the  water  has  risen  but  a  short  distance  in  the  suc- 
tion pipe.  In  the  second  diagram  the  piston  has  made  a 
sufficient  number  of  strokes  so  that  the  pump  cylinder  is 
full  of  water  which  is  flowing  out  at  the  spout  C. 

Let  hl  be  the  distance  from  the  water  level  D  to  the 
lowest  position  of  the  piston;  this  is  called  the  height  of 
lift  by  suction.  Let  h2  be  the  height  from  the  lowest  posi- 
tion of  the  piston  to  the  spout  where  the  water  flows  out; 
this  is  called  the  height  of  lift  by  the  piston.  The  distance 
hl  +  h2  is  the  vertical  height  through  which  the  water  is 
raised,  and  if  W  be  the  weight  of  water  raised  in  one  second, 
the  useful  work  per  second  is  W(ht+hJ.  The  energy  im- 
parted to  the  pump  through  the  piston  rod  is  always  greater 
than  this  useful  work,  since  energy  is  required  to  overcome 
the. f fictional  resistances  due  to  the  motion  of  the  water  and 
piston,  as  also  to  overcome  the  resistance  of  inertia  in  put- 
ting them  into  motion. 

To  discuss  the  action  of  the  pump  in  detail,  let  I  be  the 


500  PUMPS  AND  PUMPING  CHAP,  xvi 

stroke  of  the  piston,  that  is,  the  distance  between  its  highest 
and  lowest  positions.  Let  A  be  the  area  of  the  cross-sec- 
tion of  the  pump  cylinder  and  a  that  of  the  suction  pipe. 
Let  the  piston  be  supposed  to  be  at  its  lowest  position  at 
the  beginning  of  the  operation  when  no  water  has  been 
raised  in  the  suction  pipe  above  the  level  D  in  Fig.  184. 
On  raising  the  piston  through  the  stroke  /  it  describes  the 
volume  Al,  and  the  volume  of  air  ah^  now  has  the  volume 
Al  +  aki.  Let  ha  be  the  height  of  the  water  barometer 
corresponding  to  the  temperature  and  atmospheric  pres- 
sure ;  then  since  by  Mariotte's  law  the  pressure  of  a  given 
quantity  of  air  is  inversely  as  its  volume,  the  pressure-head 
h x  corresponding  to  the  volume  Al  +  aht  is  ha .  ahl/(Al  +  ah^, 
and  the  height  x  to  which  the  water  will  rise  in  the  suction 
tube  in  order  to  maintain  equilibrium  is  ha  —  h  x,  whence 

Al  Ik. 

x=h 


l  +  h^a/A) 

For  example,  let  A  be  6  and  a  be  3  square  inches,  \  be  20 
and  /  be  i  foot;  then,  under  ideal  conditions,  in  the  first 
upward  stroke  of  the  piston  the  water  rises  to  the  height 
#  =  34X0.09091=3.09  feet,  and  the  air  above  the  new 
water  level  now  has  the  normal  atmospheric  pressure. 
For  a  second  upward  stroke  of  the  piston  the  data  are  the 
same  as  before  except  that  hl  is  20—3.09  =16.91  feet,  and 
at  the  end  of  the  stroke  the  water  has  risen  a  further  dis- 
tance #  =  34X0.1058=3.60  feet,  so  that  its  surface  stands 
at  the  height  of  16.91—3.60  =  13.31  feet  below  the  lowest 
limit  of  the  piston.  Proceeding  in  like  manner  it  is  found 
that  after  the  third  upward  stroke  the  water  level  is  8.84 
feet  below  the  top  of  the  suction  pipe,  and  at  the  end  of 
the  fourth  upward  stroke  only  2.57  feet  below  it.  During 
the  fifth  upward  stroke  the  water  enters  the  pump  cylinder, 
during  the  next  downward,  stroke  it  flows  through  the 
piston  valve,  and  in  the  sixth  upward  stroke  the  water 
above  the  piston  is  lifted  and  flows  out  of  the  spout. 


ART.  184  RAISING    WATER    BY    SUCTION  501 

The  preceding  discussion  supposes  that  there  is  no 
leakage  of  air  through  and  around  the  piston,  but  this  can- 
not be  attained  in  practice ;  hence  the  degree  of  rarefaction 
below  the  piston  is  never  so  great  as  the  above  formula 
gives,  and  the  number  of  strokes  required  to  elevate  the 
water  above  the  valve  B  is  larger  than  the  computed  num- 
ber. When  the  suction  height  is  greater  than  25  feet,  it 
becomes  difficult  to  secure  sufficient  rarefaction  to  lift  the 
water  and  hence  a  foot  valve,  also  opening  upward,  is 
placed  in  the  suction  pipe  below  the  water  level  D.  The 
pump  cylinder  and  suction  pipe  can  then  be  primed,  or 
filled  with  water  from  above,  and  after  this  is  done  there 
will  be  no  difficulty  in  operating  the  pump.  If  there  be 
no  foot  valve  it  will  be  necessary  to  have  a  very  long  piston 
stroke  in  order  to  start  the  pump,  but  with  a  foot  valve 
the  stroke  may  be  any  convenient  length. 

The  action  of  this  pump  is  intermittent  and  water  flows 
from  the  spout  only  during  the  upward  stroke  of  the  pis- 
ton. When  there  are  N  upward  strokes  per  minute  the 
discharge  in  one  minute  is  NAl,  if  the  piston  and  its  valve 
be  tight.  The  useful  work  per  minute  is  NwAl(hi+h$t 
if  w  be  the  weight  of  a  cubic  unit  of  water.  When  /  and 
hi  +  hi  are  in  feet,  A  in  square  feet,  and  w  in  pounds  per 
cubic  foot,  the  horse-power  expended  in  this  useful  work  is 

ooo 

and  to  this  must  be  added  the  horse-power  required  to 
overcome  the  resistances  of  friction  and  inertia.  This 
additional  power  often  amounts  to  as  much  as  that  needed 
for  the  useful  work  and  in  this  case  the  efficiency  of  the 
pump  is  50  percent.  Suction  and  lift  pumps  are  of  numer- 
ous styles  and  sizes,  the  simplest  being  of  square  wooden 
tubes  or  of  round  tin-plate  tubes  with  leather  valves,  and 
these  can  be  readily  made  by  a  carpenter  or  tinsmith. 
They  are  mainly  used  for  small  quantities  of  water  and 
for  temporary  purposes. 


502 


PUMPS  AND  PUMPING 


CHAP.  XVI 


Prob.  184.  The  diameter  of  the  pump  cylinder  As  8  inches 
and  that  of  the  suction  pipe  is  6  inches,  while  the  vertical  dis- 
tance from  the  water  level  to  the  spout  is  23  feet.  If. the  pump 
piston  makes  30  upward  strokes  per  minute,  each  9  inches  long, 
what  horse-power  is  required  to  operate  the  pump  if  its  efficiency 
is  45  percent  ? 

ART.  185.     THE  FORCE  PUMP 

A  force  pump  is  one  that  has  a  solid  piston  which  can 
transmit  to  the  water  the  pressure  exerted  by  the  piston  rod 
and  thus  cause  it  to  rise  in  a  pipe.  The  early  force  pumps 
had  little  or  no  suction  lift,  as  the  pump  cylinder  was 
immersed  in  the  body  of  water  which  furnished  the  supply, 

but  the  modern  forms  usually 
operate  both  by  suction  and 
pressure,  the  former  occur- 
ring in  a  suction  pipe  and  the 
latter  in  the  pump  cylinder. 
Fig.  185a  shows  the  princi- 
ple of  action  of  the  common 
vertical  single-acting  suction 
and  force  pump  in  which 
there  is  no  water  above  the 
piston.  In  the  left-hand  dia- 
gram the  piston  is  ascending, 
and  the  water  is  rising  in  the 
suction  pipe  BD  under  the 
FIG.  I85a  upward  atmospheric  pres- 

sure ;  this  ascent  of  the  water  occurs  in  exactly  the  same 
manner  as  explained  in  Art.  184,  and  after  several  strokes 
its  level  rises  above  the  suction  valve  B.  The  right-hand 
diagram  shows  the  piston  descending  and  forcing  the  water 
up  the  discharge  pipe  CE.  At  C,  where  this  pipe  joins  the 
pump  cylinder,  is  a  check  valve  which  closes  on  the  upward 
stroke  and  thus  prevents  the  water  in  CE  from  returning 
into  the  pump  cylinder,  while  it  opens  on  the  downward 
stroke  under  the  pressure  of  the  water. 


B 


ART.  185 


THE  FORCE  PUMP 


503 


Let  A  be  the  area  of  the  cross-section  of  the  pump 
cylinder  and  /  the  length  of  the  stroke  of  the  piston.  Then 
at  each  upward  stroke  a  volume  of  water  equal  to  Al  is 
raised  through  the  suction  pipe  and  in  each  downward 
stroke  the  same  volume  is  raised  in  the  discharge  pipe. 
If  h  be  the  total  lift  above  the  water  level  D  and  w  the 
weight  of  a  cubic  unit  of  water,  the  work  done  in  each 
double  stroke  is  wAlh.  If  there  be  made  N  double  strokes 
per  minute,  the  useful  work  per  minute  is  NwAlh.  When 
all  dimensions  are  in  feet,  the  horse-power  required  to  do 
this  useful  work  is  found  by  dividing  this  quantity  by 
33  ooo,  and  the  actual  horse-power  required  to  run  the 
pump  is  greater  than  this  by  the  amount  needed  to  over- 
come the  f fictional  resistances.  This  additional  power  will 
depend  upon  the  length  of  the  suction  and  discharge  pipes, 
the  speed  at  which  the  pump  is  operated,  the  friction  along 
the  sides  of  the  piston,  the  losses  of  head  in  the  passage 
of  the  water  through  the  valve  openings,  and  the  losses  of 
energy  due  to  putting  the  water  into  motion  at  each  stroke. 
The  efficiency  of  single-acting  suction  and  lift  pumps  hence 
varies  between  wide  limits,  90  percent  or  more  being  ob- 
tained only  for  very  low  speeds  and  lifts,  while  for  high 
speeds  and  lifts  it  may  be  20  percent  or  less. 


FIG.  1856  FIG.  185<; 

The  cylinder  of  the  single-acting  pump  may  be  placed 
horizontal,  as  seen  in  Fig.  1856,  where  BD  is  the  suction 
piston  and  CE  the  discharge  pipe.  When  the  piston  moves 


504  PUMPS  AND  PUMPING  CHAP,  xvi 

toward  the  left,  the  suction  valve  B  opens  and  the  check 
valve  C  closes;  when  it  moves  toward  the  right,  B  closes 
and  C  opens.  The  discharge  is  intermittent,  as  in  the 
previous  case,  but  the  horizontal  position  of  the  piston 
sometimes  renders  the  connection  of  the  piston  rod  to  the 
motor  more  convenient.  If  the  height  of  the  suction  lift 
be  equal  to  that  of  the  discharge  lift,  the  force  required  to 
move  the  piston  will  be  the  same  in  each  stroke  and  the 
pump  will  work  with  less  shock  than  where  the  two  lifts 
are  unequal.  Usually,  however,  the  height  of  the  dis- 
charge lift  is  greater  than  that  of  the  suction  lift  and  the 
force  required  to  move  the  piston  is  then  the  greatest  when 
it  moves  from  left  to  right  in  Fig.  1856.  In  order  to  equalize 
the  forces  exerted  by  the  motor  the  duplex  pump  was  de- 
vised; this  consists  of  two  single-acting  cylinders  placed 
side  by  side  and  connected  to  the  same  suction  and  dis- 
charge pipe,  the  pistons  moving  so  that  one  exerts  suction 
while  the  other  is  forcing  the  water  upward.  Three  single- 
acting  cylinders  are  also  sometimes  connected  with  the 
same  suction  and  discharge  pipe,  in  which  case  it  is  called 
the  triplex  pump.  Duplex  and  triplex  pumps  give  a 
continuous  flow  of  water  in  both  the  suction  and  discharge 
pipes  and  thus  diminish  the  shocks  that  occur  in  a  pump 
with  one  cylinder,  while  the  efficiency  is  materially  in- 
creased because  the  losses  due  to  starting  and  stopping 
the  columns  of  water  are  in  large  part  avoided. 

A  double-acting  pump  is  one  having  a  single  cylinder 
in  which  a  solid  piston  or  plunger  exerts  suction  and  pres- 
sure in  both  strokes  and  thus  gives  a  nearly  continuous 
flow  through  suction  and  discharge  pipes.  Fig.  185d  shows 
the  form  known  as  the  piston  pump,  while  Fig.  185e  is 
that  called  the  plunger  pump,  the  piston  being  replaced 
by  a  long  cylinder  moving  in  a  short  stuffing  box  AA.  In 
both  figures  D  is  the  suction  pipe  and  E  the  discharge  pipe. 
When  the  piston  moves  from  left  to  right,  the  valves  BL 
and  C2  open,  while  B2  and  Cl  close;  when  it  moves  in  the 


ART.  185 


THE  FORCE  PUMP 


505 


opposite  direction  B2  and  C1  open,  while  B1  and  C2  close. 
The  plunger  pump  was  invented  in  the  seventeenth  cen- 
tury, and  its  advantages  over  the  piston  type  are  so 
great  that  it  is  now  extensively  used  for  large  pumping 
machinery.  The  cylinder  of  the  piston  pump  must  be 
bored  to  an  exact  and  uniform  size  and  its  piston  must  be 


FIG.  185d 


carefully  packed,  while  in  the  plunger  pump  only  the  short 
length  of  the  stuffing  box  is  bored  and  packed,  the  plunger 
itself  having  no  packing.  The  water  lifted  in  one  stroke 
of  either  pump  is  Al,  where  A  is  the  area  of  the  piston  and 
/  the  length  of  its  stroke,  provided  there  is  no  leakage  past 
the  packing. 

For  all  these  forms  of  pumps  a  foot  valve  should  be 
placed  in  the  suction  pipe,  if  the  suction  lift  exceeds  20 
feet,  in  order  that  the  pump  may  be  readily  primed  (Art. 
184).  To  reduce  the  shocks  that  occur  to  a  certain  extent 
even  in  the  double-acting  pumps,  an  air  chamber  is  fre- 
quently attached  to  the  discharge  pipe  so  that  the  con- 
fined air  may  distribute  and  lessen  the  shock  that  would 
otherwise  be  concentrated  on  the  end  of  the  discharge  pipe. 
Fig.  185c  shows  such  an  air  chamber  attached  to  a  single- 
acting  pump ;  in  the  upper  part  of  it  is  seen  the  compressed 
air  which  is  receiving  the  pressure  from  the  piston.  After 
the  check  valve  C  closes  the  pressure  of  this  air  maintains 
the  flow  up  the  discharge  pipe  E  and  hence  the  air  chamber 


506  PUMPS  AND  PUMPING  CHAP,  xvi 

helps  to  avoid  the  losses  due  to  intermittent  flow.  A 
duplex  pump  or  a  double-acting  pump,  when  provided 
with  an  air  chamber  of  proper  size,  will  work  very  smoothly. 

Prob.  185.  Consult  Ewbanks'  Hydraulics  and  Mechanics 
(New  York,  1847)  and  describe  a  method  of  raising  water  through 
a  low  lift  by  means  of  a  frictionless  plunger  pump.  Ewbank 
notes  that  a  stout  young  man  weighing  134  pounds  raised  8J 
cubic  feet  per  minute  with  this  machine  to  a  height  of  1 1 J  feet 
and  worked  at  this  rate  nine  hours  per  day.  If  the  efficiency  of 
this  pump  was  unity,  what  horse-power  did  the  stout  young  man 
exert?  Was  his  performance  high  or  low? 


ART.  186.     LOSSES  IN  THE  FORCE  PUMP 

A  reliable  numerical  computation  of  the  hydraulic 
losses  of  energy  in  the  force  pump  cannot  be  made  without 
knowing  the  constants  to  use  in  finding  the  losses  of  head 
due  to  the  valves  (Art.  88),  and  these  have  been  experi- 
mentally determined  for  only  a  few  special  forms.  The 
valves  shown  in  most  of  the  figures  of  the  preceding  articles 
are  simple  flap  valves,  but  puppet  valves  are  more  gen- 
erally used,  and  Fig.  185^  indicates  such.  In  passing 
through  a  valve  the  water  loses  energy  in  friction,  and  also 
in  impact  due  to  the  subsequent  expansion.  Since  pumps  are 
made  in  numerous  forms  having  different  details,  general 
discussions  of  losses  are  difficult  to  make.  The  attempt  will, 
however,  be  undertaken  for  the  plunger  force  pump  of 
Fig.  185^.  Let  h  be  the  total  height  through  which  the 
water  is  lifted  by  both  suction  and  pressure,  and  hf  be  the 
sum  of  all  the  hydraulic  losses  of  head.  Let  K  be  the 
energy  delivered  per  second  to  the  piston  rod,  k'  the  energy 
expended  in  friction  in  the  stuffing  boxes  of  the  piston  rod 
and  plunger,  q  the  discharge  per  second,  and  w  the  weight 
of  a  cubic  unit  of  water.  Then 


ART.  188  LOSSES    IN    THE    FORCE    PUMP  507 

and  the  pump  should  be  so  arranged  as  to  make  the  losses 
k'  and  hf  as  small  as  possible.  Only  the  hydraulic  losses 
will  be  considered  in  the  following  discussion. 

By  means  of  the  principles  of  Chapter  VII  a  rough  for- 
mulation of  the  elements  that  make  up  the  lost  head  h'  can 
be  effected,  supposing  the  flow  in  the  pipes  to  be  steady. 
Let  /!  be  the  length,  d1  the  diameter,  and  vl  the  velocity  for 
the  suction  pipe,  and  /2,  d2,  and  v2  the  same  things  for  the 
discharge  pipes.  Let  2H  be  the  number  of  valves  in  the  suc- 
tion and  discharge  chambers  (Fig.  1850),  all  being  taken  of 
the  same  size,  and  let  V  denote  the  velocity  of  the  water 
through  each  valve  opening.  Let  these  chambers  be  so 
large  that  the  velocity  of  the  water  through  them  is  very 
small  compared  to  that  in  the  pipes  and  valve  openings. 
Then 

g+f^        (186) 

gives  all  the  hydraulic  losses  of  head.  In  the  first  paren- 
thesis m  indicates  the  loss  due  to  entrance  at  the  foot  of 
the  suction  pipe  (Art.  85),  fl^/d^  the  friction  loss  in  the  suc- 
tion pipe  (Art.  86),  and  i  the  loss  due  to  expansion  (Art.  74) 
as  the  water  enters  the  suction  chamber  B1B2.  In  the 
second  parenthesis  nm  indicates  the  loss  due  to  the  open 
valves  (Art.  88)  and  n  that  due  to  sudden  expansion  as  the 
water  enters  the-  pump  cylinder  through  the  suction  valves 
and  the  discharge  chamber  C\C2  through  the  discharge 
valves.  The  last  term  gives  the  loss  due  to  friction  in  the 
discharge  pipe.  If  there  is  an  air  chamber  on  the  discharge 
pipe  another  term  might  be  introduced,  but  as  the  effect 
of  the  air  chamber  in  reducing  water  hammer  is  a  beneficial 
one,  this  term  need  not  be  used.  The  starting  and  stopping 
of  the  piston  brings  in  other  losses  of  energy,  but  as  these 
are  not  hydraulic  losses  they  will  not  be  considered  here. 

When  the  pipes  are  long  the  losses  due  to  pipe  fric- 
tion will  far  exceed  those  in  the  pump  and  are  not  fni  1v 


508  PUMPS  AND  PUMPING  CHAP,  xvi 

chargeable  against  it  as  a  machine;  hence  to  consider  the 
pump  alone  the  lengths  l^  and  12  may  be  made  equal  to  zero, 
as  also  m  in  the  first  parenthesis.  Then  (186)  becomes 

v  2  V2 

h'  =—  +(nm 


in  which  the  first  term  of  the  second  member  gives  the  loss 
of  head  in  entering  the  suction  chamber,  and  the  second 
those  occurring  in  entering  and  leaving  the  pump  cylinder. 
This  equation  appears,  at  first  thought,  to  indicate  that  a 
suction  chamber  is  not  a  hydraulic  advantage,  although  it 
is  known  to  afford  a  practical  advantage  in  causing  the 
valves  to  operate  successfully,  as  also  in  permitting  ready 
access  to  them.  If  a  be  the  area  of  each  valve  opening,  and 
ax  that  of  the  suction  pipe,  then  a^  must  equal  \ndV  ,  since 
the  same  quantity  of  water  passes  per  second  through  the 
suction  pipe  and"  through  \n  valves.  Accordingly  the  total 
loss  of  head  in  the  pump  may  be  written 


na 

which  clearly  shows  that  this  loss  decreases  as  the  number  of 
valves  increases.  But  the  number  of  valves  cannot  con- 
veniently be  made  greater  than  four  without  using  the  suc- 
tion and  discharge  chambers ;  such  chambers  may  therefore 
be  made  to  give  a  real  hydraulic  advantage  by  using  8,  12, 
or  1 6  valves- and  making  the  area  of  each  valve  opening 
sufficiently  large. 

As  a  numerical  example,  take  a  plunger  force  pump,  like 
Fig.  1850,  having  a  piston  area  A  =  0.84  square  feet  and  a 
stroke  of  1.25  feet,  the  number  of  single  strokes  per  minute 
being  30.  The  volume  of  water  lifted  per  second  is  hence 
30  Xo.82  x  i. 25/60  =0.525  cubic  feet.  Let  the  diameter  of 
the  suction  pipe  be  10  inches  and  the  area  of  its  cross-section 
aL=  0.545  square  feet.  The  mean  velocity  in  the  suction 
pipe  is  then  0.5  2  5/0. 5 45  =0.96  feet  per  second.  Let  there  be 
12  valves  in  the  suction  chamber,  so  that  n  =6,  and  let  the 


ART.  186  LOSSES    IN    THE   FORCE    PUMP  509 

area  of  each  valve  opening  be  a  =  8  square  inches  =0.05  5 6 
square  feet.  The  velocity  through  each  of  the  open  valves 
is  then  1^=0.525/3  X 0.05 56  =3. 15  feet  per  second.  As  Art. 
88  does  not  give  the  values  of  m  for  puppet  valves,  it  may  be 
here  noted  that  the  experiments  of  Bach*  indicate  that 
they  range  from  i.i  to  2.8,  depending  upon  the  height  of 
valve  lift  and  the  width  of  the  seat.  Taking  2  as  a  mean 
value  of  m,  the  lost  head  in  the  pump  is 

*' -0.01555(1 +3  xf(o^)')o.96>-2.77  feet. 

The  useful  head  h,  when  the  lengths  of  the  suction  and  dis- 
charge pipes  are  disregarded,  is  probably  about  3  feet,  so 
that  the  hydraulic  efficiency  is  e=h/(h  +  hr)  =0.52.  If  the 
lengths  of  the  vertical  suction  and  discharge  pipes  be  each 
20  feet  and  their  diameters  be  10  inches,  the  useful  head  h 
is  about  43  feet  and  from  (186)  the  value  of  h'  is  found  to 
be  about  5.8  feet,  so  that  the  hydraulic  efficiency  is  about 
0.88.  The  velocity-head  v.?/2g  which  is  lost  at  the  top  of 
the  discharge  pipe  is  here  only  o.oi  feet,  so  that  it  is  unneces- 
sary to  consider  it  in  determining  the  efficiency. 

This  discussion  shows  that  the  losses  of  head  in  force 
pumps  may  be  made  very  slight  by  running  them  at  low 
speeds  in  order  that  the  velocity  vl  may  be  small.  It  shows 
that  the  losses  decrease  as  the  areas  of  the  valve  openings 
and  their  number  are  increased.  It  shows  that,  for  vertical 
suction  and  discharge  pipes,  the  efficiency  increases  with 
the  useful  lift  h,  if  the  velocity  in  the  pipes  is  the  same  for 
different  lifts.  These  conclusions  are  verified  by  experi- 
ments, some  of  which  will  be  noted  in  the  next  article. 
Since  the  flow  through  the  valves  and  pump  cylinder  is  not 
quite  steady,  numerical  computations  like  the  above  cannot, 
however,  be  expected  to  give  more  than  rough  approximate 
results ;  nevertheless  such  results  are  useful  in  indicating  the 
influence  of  the  resistances  upon  the  efficiency. 

*  Zeitschrift  deutscher  Ingenieur  Verein,  1886,  p.  421. 


510  PUMPS  AND  PUMPING  CHAP,  xvi 

Prob.  186.  For  the  above  numerical  example,  compute  the 
horse-power  required  to  run  the  pump  when  the  useful  lift  is 
43  feet,  assuming  that  3  percent  of  that  power  is  expended  in 
overcoming  friction  in  the  stuffing  boxes. 


ART.  187.      PUMPING  ENGINES 

The  steam  engine  was  invented  and  perfected  through 
the  desire  to  devise  methods  of  pumping  water  better  than 
those  in  which  the  power  of  men  and  horses  were  used. 
Worcester  in  1633,  and  Papin  in  1695,  used  the  direct  pres- 
sure of  steam  upon  water  in  a  cylinder,  and  Savery  in  1700 
used  both  such  pressure  and  the  partial  vacuum  caused  by 
the  condensation  of  the  steam.  Newcomen  in  1705  used 
a  piston  on  one  side  of  which  steam  was  applied  and  con- 
densed, the  motion  of  the  piston  being  communicated  by  a 
walking  beam  to  the  piston  rod  of  a  pump.  Watt,  about 
1775,  introduced  the  crank,  the  parallel  motion,  the  cut-off, 
the  governor,  and  other  improvements ;  he  also  brought  the 
steam  on  both  sides  of  the  piston,  thus  making  the  engine 
double-acting.  The  first  important  application  of  the  steam 
engine  was  in  operating  pumps  to  drain  mines,  but  it  soon 
came  into  use  in  all  branches  of  industry  where  power  was 
needed.  Its  influence  on  modern  progress  has  been  great. 

The  modern  pumping  engine  consists  of  one  or  more 
steam  cylinders  connected  to  the  same  number  of  pump 
cylinders  by  piston  rods,  so  that  the  steam  pressure  is  directly 
transmitted  through  them  to  the  water.  It  is  important 
that  the  pressure  in  the  water  cylinder  should  be  maintained 
nearly  constant  during  the  length  of  the  stroke  and  hence 
the  steam  should  not  be  used  expansively  in  the  usual  way ; 
to  insure  constant  steam  pressure  some  form  of  compensator 
is  used.  The  water  cylinders  are  usually  of  the  plunger 
type,  and  these  are  connected  to  the  same  suction  and  dis- 
charge pipes,  an  air  chamber  being  placed  on  the  latter  to 
relieve  the  pump  chambers  of  shock  and  ensure  steady  flow.. 


ART.  187  PUMPING   ENGINES  511 

The  boilers,  steam  cylinders,  and  water  cylinders  constitute 
one  machine  or  apparatus  called  a  pumping  engine.  The 
efficiency  of  this  apparatus  is  low,  for  it  is  equal  to  the 
product  of  the  efficiencies  of  its  separate  parts.  The  effi- 
ciency of  the  furnace  and  boiler  is  about  75  percent  in  the 
best  designs,  the  efficiency  of  the  steam  cylinders  about  30 
percent,  and  that  of  the  water  cylinders  about  80  percent, 
so  that  the  efficiency  of  the  pumping  engine  as  a  whole  is 
only  1 8  percent.  This  means  that  only  18  percent  of  the 
energy  of  the  fuel  is  utilized  in  lifting  the  water,  and  this, 
figure  is,  indeed,  a  high  efficiency,  for  many  pumping  plants, 
are  operated  with  an  efficiency  of  less  than  10  percent. 

The  term  ' '  duty ' '  is  often  employed  as  a  measure  of 
the  performance  of  a  pumping  engine,  instead  of  expressing 
it  by  an  efficiency  percentage.  This  term  was  devised  by 
Watt,  who  defined  duty  as  the  number  of  foot-pounds  of 
useful  work  produced  by  the  consumption  of  100  pounds, 
of  coal.  On  account  of  the  variable  quality  of  coal  a  more 
precise  definition  of  duty  was  introduced  in  1890  by  a  com- 
mittee of  the  American  Society  of  Mechanical  Engineers, 
namely,  that  duty  should  be  the  number  of  foot-pounds 
of  work  produced  by  the  expenditure  of  i  ooo  ooo  English 
thermal  heat  units.  One  English  thermal  heat  unit  is  that 
amount  of  energy  which  will  raise  one  pound  of  pure  water 
one  Fahrenheit  degree  in  temperature  when  the  water  is 
at  or  near  the  temperature  of  maximum  density  (Art.  4) ; 
this  amount  of  energy  is  778  foot-pounds,  and  this  constant 
is  called  the  mechanical  equivalent  of  heat.  The  duty  of 
a  perfect  pumping  engine,  in  which  no  losses  of  any  kind 
occur,  would  be  778  ooo  ooo  foot-pounds.  The  highest 
duty  obtained  in  a  test  is  about  160  ooo  ooo  foot-pounds 
and  the  efficiency  of  such  an  engine  is  160/778=0. 21.  Com- 
mon pumping  engines  have  duties  ranging  from  120  ooo  ooo 
to  60  ooo  ooo,  the  corresponding  efficiencies  being  from  1.5 
to  7.5  percent.  The  modern  definition  of  duty  agrees  with 
that  of  Watt,  if  the  coal  used  be  of  such  quality  that  one 


512  PUMPS  AND  PUMPING  CHAP,  xvi 

pound  of  it  possesses  a  potential  energy  of  10  ooo  English 
heat  units,  which  is  somewhat  less  than  that  obtain- 
able from  average  coal.  The  higher  the  duty  of  a  pumping 
engine  the  greater  is  the  amount  of  work  that  can  be  per- 
formed by  burning  a  given  quantity  of  coal.  A  high-duty 
engine  is  hence  economical  and  a  low-duty  engine  is  waste- 
ful in  coal  consumption,  but  the  first  cost  of  the  former  is 
much  greater  than  that  of  the  latter. 

A  duty  test  of  a  pumping  engine  consists  in  determining 
the  number  of  heat  units  furnished  by  a  given  quantity  of 
coal,  the  quantity  of  water  lifted  by  the  pump,  the  leakage 
past  the  piston  packing,  the  pressure-heads  in  the  suction 
and  discharge  pipes,  the  indicated  horse-power  of  the 
steam  cylinders,  and  many  other  minor  quantities  needed 
for  estimating  the  efficiency  of  the  boiler  and  steam  part 
of  the  apparatus.  The  usual  method  of  determining  the 
discharge  is  by  the  displacement  of  the  piston  or  plunger; 
if  A  be  the  area  of  its  cross-section,  /  the  length  of  the  stroke, 
N  the  number  of  single  strokes  during  the  test,  and  T  the 
number  of  seconds  during  which  the  test  lasted,  then  NAl 
is  the  total  quantity  of  water  lifted,  and 

q=cNAl/T 

is  the  quantity  lifted  per  second,  c  being  a  coefficient  which 
takes  account  of  the  leakage  or  slip  past  the  plunger.  The 
value  of  c  is  to  be  found  by  removing  one  of  the  cylinder 
heads  and  admitting  water  on  the  other  side  of  the  plunger, 
and  its  value  is  usually  from  0.99  to  0.95  in  new  pumps. 
The  total  pressure-head  H  is  found  from 


where  h±  and  h2  are  the  pressure-heads  corresponding  to  the 
mean  readings  of  the  gages  on  the  suction  and  discharge 
pipes  and  d  the  vertical  distance  between  the  centers  of 
the  gages  ;  here  the  plus  sign  is  to  be  used  when  the  corre- 
sponding pressure  is  below  and  the  minus  sign  when  it  is 
above  that  of  the  atmosphere.  The  total  work  done  by 


ART.  187  PUMPING    ENGINES  513 

the  pump  during  the  trial  is  then  cNAl.H  and  then  the 
duty  of  the  pumping  engine 

Duty  =  i  ooo  ooocNAlH/heat  units. 

in  which  the  denominator  is  determined  by  the  thermo- 
dynamic  tests  made  on  the  boiler  and  steam  engine.  The 
capacity  of  the  pump,  or  the  quantity  of  water  lifted  in 
24  hours,  is  24X3600X9. 

The  efficiency  of  pump  cylinders,  which  are  tested  in 
the  above  manner,  is  usually  found  by  dividing  the  work 
wqH  done  by  them  in  one  second  by  that  done  by  the  steam 
as  determined  by  indicator  cards  taken  from  the  steam 
cylinders.  This  method  differs  from  that  used  in  the 
previous  articles,  and  gives  results  too  small  from  the 
standpoint  of  hydraulic  losses.  A  discussion  by  Webber  * 
of  several  tests  shows  that  this  efficiency  increases  with 
the  lift  as  follows : 

Lift  in  feet,  5  15          30          100        170        270 

Efficiency,          0.30       0.45       0.65       0.85       0.91       0.88 

The  highest  value  of  91  percent  was  obtained  from  a  test 
of  a  Leavitt  pumping  engine  having  a  duty  of  in  549  ooo 
foot-pounds  and  a  capacity  of  4  400  ooo  gallons  per  24 
hours ;  the  duration  of  this  test  was  15.1  hours. 

Prob.  187a.  Using  coal  of  the  standard  quality,  show  that 
100  pounds  burned  in  one  hour  produces  75.8  horse-powers  with 
a  150  ooo  ooo-duty  pumping  engine. 

Prob.  1876.  In  a  test  lasting  12  hours,  27  502  ooo  heat  units 
were  produced  under  the  boiler.  The  area  of  the  plunger  was 
172  square  inches,  the  length  of  the  stroke  was  18.9  inches,  the 
number  of  single  strokes  was  76  ooo,  and  the  leakage  past  the 
plunger  packing  was  5900  cubic  feet.  The  pressure  gage  on 
the  force  pipe  read  i  oo  and  the  vacuum  gage  on  the  suction  pipe 
read  9.3  pounds  per  square  inch,  the  distance  between  the  cen- 
ters of  these  gages  being  8  feet.  The  mean  indicated  horse- 
power of  the  steam  cylinders  was  128.  Compute  the  discharge 

*  Transactions  American  Society  of  Mechanical  Engineers,    1886,   vol. 
7,  p.  602. 


514 


PUMPS  AND  PUMPING 


CHAP.  XVI 


of  the  pump  in  cubic  feet  per  second  and  its  capacity  in  gallons 
per  day.  Compute  the  total  pressure-head  H.  Compute  the 
duty  of  the  pumping  engine.  Compute  the  efficiency  of  the 
pump  cylinders. 

ART.  188.     THE  CENTRIFUGAL  PUMP 

The  centrifugal  pump  is  the  reverse  of  a  turbine  wheel 
and  any  reaction  turbine,  when  run  backwards  by  power 
applied  to  its  axle,  will  raise  water  through  its  penstock. 
The  centrifugal  pump,  like  the  turbine,  is  of  modern  origin 
and  development.  A  rude  form,  devised  by  Ledemour  in 
1730,  consisted  of  an  inclined  tube  attached  by  arms  to  a 
vertical  shaft;  the  lower  end  of  the  tube  being  immersed, 
the  water  flowed  from  its  upper  end  when  the  shaft  was 
rotated.  It  was  not,  however,  until  about  1840  that  the 
first  true  centrifugal  pumps  came  into  use,  and  they  have 
since  been  perfected  so  as  to  be  of  great  value  in  engineering 
operations,  especially  for  low  lifts. 

Fig.  188  shows  the  principle  of  the  arrangement  and 
action  of  the  centrifugal  pump.  The  power  is  applied 

through  the  axis  A 
to  rotate  the  wheel 
BB  in  the  direction 
indicated  by  the  ar- 
row. This  wheel  is 
formed  of  a  number 
of  curved  vanes  like 
those  in  a  turbine 
wheel  (Art.  165). 
The  revolving  vanes 
produce  a  partial 

FIG.  188  vacuum     and     this 

causes  the  water  to  rise  in  the  suction  pipe  DD  which  enters 
through  the  center  of  the  wheel  case  and  delivers  the  water 
at  the  axis  of  the  wheel.  The  water  is  then  forced  outward 
through  the  vanes  and  passes  into  the  volute  chamber  CCt 


••* 

B 

to       1 

B 

D 

G 

ART.  188  THE    CENTRIFUGAL    PUMP  515 

which  is  of  varying  cross-section  in  order  to  accommodate 
the  increasing  quantity  of  water  that  is  delivered  into  it 
and  all  of  which  passes  up  the  discharge  pipe  E.  The  rota- 
tion of  the  wheel  hence  produces  a  negative  pressure  at 
the  upper  end  of  the  suction  pipe  and  a  positive  pressure 
in  the  volute  chamber,  and  the  water  rises  in  the  pipes 
in  the  same  manner  as  in  those  of  a  suction  and  force 
pump.  The  height  of  the  suction  lift  cannot  usually  ex- 
ceed about  28  feet. 

The  parallelograms  of  velocity  shown  in  Fig.  188  are 
the  same  as  in  the  reaction  turbine  (Art.  165)  and  a  similar 
notation  will  be  used.  The  velocities  of  rotation  of  the 
inner  and  outer  circumferences  will  be  called  u  and  ult  the 
absolute  velocities  of  the  water  as  it  enters  and  leaves  the 
wheel  are  VQ  and  vlt  and  the  corresponding  relative  velocities 
are  V  and  Vlt  The  angles  of  entrance,  approach,  and  exit 
are  called  a,  <£,  and  /2,  while  6  denotes  the  angle  between 
vl  and  ur  Let  H0  be  the  pressure-head  at'  the  top  of  the 
entrance  pipe  and  Hl  that  at  the  foot  of  the  discharge  pipe, 
while  h0  and  h^  are  the  heights  of  the  suction  and  force  lifts 
estimated  downward  and  upward  from  the  center  of  the 
wheel,  and  let  ha  be  the  height  of  the  water  barometer. 
Then  from  (153) 


and  also  from  (32)  2,  not  considering  f  fictional  resistances, 


. 

Combining  these  equations,  and  replacing  h^  +  h^  by  h, 
where  h  is  the  total  lift,  the  fundamental  equation  for  the 
discussion  of  frictionless  centrifugal  pumps  results.  To 
introduce  the  f  fictional  losses,  however,  h  +  h'  should  be 
used  instead  of  h,  where  h'  is  the  total  head  lost  in  all  the 
liydraulic  resistances.  Then 


516  PUMPS  AND  PUMPING  CHAP,  xvr 

is  the  formula  applicable  to  actual  cases.  If  q  be  the  dis- 
charge per  second  and  w  the  weight  of  a  cubic  unit  of  water, 
the  work  of  the  pump  per  second,  not  including  axle  friction, 


A  centrifugal  pump,  like  a  reaction  turbine,  must  be  run 
at  a  certain  speed  in  order  to  give  the  maximum  efficiency, 
and  this  is  now  to  be  determined.  In  the  first  place,  it  must 
be  noted  that  the  water  enters  the  vanes  radially,  since 
there  are  no  guides  ;  hence  the  entrance  angle  a  must  be 
90°,  and  accordingly  V2=u2+vQ2.  Secondly,  the  parallel- 
ogram of  velocities  at  exit  gives  V^  =  u12  +  vl*  —  2Uivl  cos  0. 
Introducing  these  conditions  into  (188)  ,  it  reduces  to 

UM  cosd=g(h  +  h')  (188), 

Now,  in  order  that  the  water  may  enter  the  volute  chamber 
with  as  little  loss  in  impact  as  possible,  the  velocity  vt  cos# 
should  be  the  same  as  that  in  the  volute  chamber  and  the 
latter  should  be  the  same  as  that  in  the  discharge  pipe.  Let 
q  be  the  discharge  per  second  and  a  the  area  of  the  cross- 
section  of  that  pipe,  then  vt  cos#  should  equal  q/at  and  con- 
sequently 


is  the  advantageous  velocity  of  the  outer  circumference  of 
the  wheel.  For  a  given  discharge  q  the  advantageous  speed 
of  the  centrifugal  pump  increases  directly  as  the  total  pres- 
sure-head h  +  h';  for  a  given  head  h  +  hf  the  speed  varies  in- 
versely as  the  discharge  q. 

Since  the  speed  must  increase  with  the  lift,  and  since  the 
losses  of  head  increase  with  the  speed,  it  follows  that  the  effi- 
ciency of  the  centrifugal  pump  in  general  decreases  with  the 
lift.  This  theoretic  conclusion  has  been  verified  by  prac- 
tical tests.  Webber,  in  his  discussion  cited  in  the  last  article, 
gives  the  following  as  the  mean  results  derived  from  a  num- 
ber of  experiments,  the  efficiency  computed  being  the  ratio 
of  the  work  done  by  the  pump  to  that  obtained  from  indi- 
cator cards  taken  on  the  cylinders  of  the  steam  motor  : 


ART.  189  THE    HYDRAULIC   RAM  517 

Lift  in  feet,  5  10          20          40          60 

Efficiency,          0.56       0.64       0.68       0.58       0.40 

For  a  low  lift  the  centrifugal  pump  has  a  hydraulic  efficiency 
higher  than  these  figures  indicate,  but,  as  in  the  case  of  the 
force  pump,  it  is  difficult  to  determine  reliable  values  by 
numerical  computations. 

The  centrifugal  pump  possesses  an  advantage  over  the 
force  pump  in  having  no  valves  and  in  being  able  to  handle 
muddy  water,  for  even  gravel  may  pass  through  the  vanes 
without  injuring  them.  The  above  figure  represents  the 
principle  rather  than  the  actual  details  of  construction. 
Usually  the  suction  pipe  is  divided  into  two  parts  which 
enter  the  axis  upon  opposite  sides  of  the  wheel,  and  the 
volute  chamber  is  often  made  wider  than  the  wheel  case, 
thus  forming  what  is  called  a  whirlpool  chamber,  which  pre- 
vents some  of  the  losses  of  head  due  to  impact.  The  vanes 
are  sometimes  curved  in  the  opposite  direction  to  that  shown 
in  the  figure,  as  by  so  doing  the  angle  6  is  made  smaller  and 
the  speed  of  the  pump  is  lessened,  as  is  seen  from  formula 
(188)  2-  The  theory  of  the  centrifugal  pump  is,  however, 
much  less  definite  than  that  of  the  reaction  turbine,  and 
experiment  is  the  best  guide  to  determine  the  advantageous 
shape  of  the  vanes. 

Prob.  188.  A  centrifugal  pump  lifts  120  cubic  feet  of  water 
per  minute  through  a  discharge  pipe  having  a  diameter  of  i 
foot.  The  outer  diameter  of  the  wheel  is  2  feet  and  the  num- 
ber of  revolutions  per  second  is  60.  If  the  height  through 
which  the  water  is  lifted  is  20  feet,  show  that  the  hydraulic 
efficiency  is  about  67  percent. 

ART.  189.      THE  HYDRAULIC  RAM 

The  hydraulic  ram  is  an  apparatus  which  employs  the 
dynamic  pressure  produced  by  stopping  a  column  of  moving 
water  to  raise  a  part  of  this  water  to  a  higher  level  than  that 
of  its  source.  The  principle  of  its  action  was  recognized  by 
Whitehurst  in  1772,*  but  the  credit  of  perfecting  the  ma- 

*  Transactions  Royal  Society,  1775,  vol.  65,  p.  277. 


518  PUMPS  AND  PUMPING  CHAP,  xvi 

chine  is  due  to  Montgolfier,  who  in  1796  built  the  first  self- 
acting  ram.  It  has  since  been  widely  used  for  pumping 
small  quantities  of  water  from  streams  to  houses,  but  is  not 
so  well  adapted  to  lifting  a  large  quantity;  many  attempts 
have  been  made  in  this  direction,  some  of  which  give  promise, 
of  much  usefulness. 

The  principle  of  the  action  of  the  hydraulic  ram  is  shown 
in  Fig.  189,  where  A  is  the  reservoir  that  furnishes  the  supply, 


FIG.  189 

BCD  the  ram,  A B  the  drive  pipe  which  carries  the  water  to 
the  ram,  DE  the  discharge  pipe  through  which  a  part  of  the 
water  is  raised  to  the  tank  E.  The  ram  itself  consists 
merely  of  the  waste  valve  B  through  which  a  part  of  the 
water  from  the  drive  pipe  escapes,  and  the  air  vessel  D 
which  has  a  valve  C  that  allows  water  to  enter  it  through 
BC,  but  prevents  its  return.  The  waste  valve  B  is  either 
weighted  or  arranged  with  a  spring  so  that  it  will  open  when 
acted  upon  by  the  static  pressure  due  to  the  head  H.  As 
soon  as  it  opens  the  water  flows  through  it,  but  as  the  velocity 
increases  the  dynamic  pressure  due  to  the  motion  of  the 
column  AB  (Art.  148)  becomes  sufficiently  great  to  close 
the  valve  B.  Then  this  dynamic  pressure  opens  the  valve 
C  and  compresses  the  air  in  the  air  chamber  or  forces  water 
up  the  discharge  pipe.  A  moment  later  when  equilibrium 
has  obtained  in  the  air  vessel  the  valve  C  closes  and  the  air 
pressure  maintains  the  flow  for  a  short  period  in  the  discharge 
pipe  while  the  water  in  the  drive  pipe  comes  to  rest.  Then 
the  waste  valve  B  opens  again  and  the  same  operations  are 
repeated. 


ART.  189  THE   HYDRAULIC    RAM  519 

The  algebraic  discussion  of  the  hydraulic  ram  is  very  dif- 
ficult because  it  involves  the  time  in  which  the  waste  valve 
closes  and  the  law  of  its  rate  of  closing.  The  investigation 
in  Art.  148,  however,  clearly  shows  that  the  operations 
above  described  will  take  place  if  the  drive  pipe  is  long 
enough  to  produce  a  dynamic  pressure  sufficient  to  close 
the  waste  valve.  Let  /  be  the  length  of  that  pipe,  v  the 
velocity  in  it,  p0  the  static  unit  --pressure  due  to  H,  w  the 
weight  of  a  cubic  unit  of  water,  g  the  acceleration  of  gravity, 
and  t  the  time  in  which  the  valve  closes.  Then,  since  there 
is  no  static  pressure  at  the  valve  during  the  flow,  (148)j  gives 

p  =  2Wlv/gt-pQ 

which  is  a  good  approximation  to  the  excess  of  dynamic 
pressure  over  the  static  pressure  p0.  It  is  seen  that  this  ex- 
cess p  may  be  rendered  very  great  by  making  /  large  and  t 
small,  and  that  its  greatest  value  is 

p=wuv/g-pQ 

in  which  u  is  the  velocity  of  sound  in  water.  It  is  rare,  how- 
ever, that  a  drive  pipe  is  sufficiently  long  to  furnish  the  ex- 
cess dynamic  pressure  given  by  the  last  formula. 

The  efficiency  of  the  hydraulic  ram  is  the  ratio  of  the 
useful  work  done  to  the  energy  expended  in  the  waste  water. 
Let  q  be  the  quantity  of  water  lifted  per  second  through  the 
height  h  from  the  level  of  the  reservoir  A  to  that  of  the  tank 
E.  Let  Q  be  the  discharge  per  second  through  the  waste 
valve  and  H  the  height  through  which  it  falls,  then  the  effi- 
ciency of  the  ram  and  its  pipes  is 

wqh        qh 

<i/ii/~)  LJ        /")  T  T 
UU\)L~L       \J r~L 

It  is  found  by  experiment  that  the  efficiency  decreases  as 
the  ratio  h/H  increases.  Eytelwein  found  that  e  was  0.92 
when  h/H  was  unity,  0.67  when  h/H  was  5,  and  0.23  when 
h/H  was  20,  but  these  values  were  probably  derived  by 
using  a  different  formula  for  the  efficiency. 


520  PUMPS  AND  PUMPING  CHAP,  xvi 

Experiments  in  1890  at  Lehigh  University  on  a  Gould 
ram  No.  2,  in  which  the  waste  valve  made  55  strokes  per 
minute,  gave  a  mean  efficiency  of  35  percent.  The  length 
of  the  supply  pipe  was  38  feet  and  its  fall  12  feet,  the  length 
of  the  discharge  pipe  60  feet,  and  the  lift  h  was  12  feet,  so 
that  the  ratio  h/H  was  unity.  These  experiments  showed 
also  that  the  efficiency  increased  as  the  number  of  strokes 
per  minute  was  decreased  by  lessening  the  weight  on  the 
waste  valve.  The  maximum  quantity  of  water  raised  per 
minute,  however,  occurred  with  a  heavier  waste  valve  than 
that  which  gave  the  maximum  efficiency.  The  efficiency 
was  also  found  to  increase  as  the  length  of  the  stroke  of  the 
waste  valve  decreased. 

The  least  possible  fall  in  the  drive  pipe  of  the  hydraulic 
ram  is  about  i  J  feet  and  the  least  length  of  drive  pipe  about 
1 5  feet.  It  is  customary  to  make  the  area  of  the  discharge 
pipe  from  one-third  to  one-fourth  that  of  the  drive  pipe, 
and  with  these  proportions  a  fall  of  10  feet  will  force  water 
to  a  height  of  nearly  150  feet.  A  common  rule  of  manufac- 
turers is  that  about  one-seventh  of  the  water  flowing  down 
the  drive  pipe  may  be  raised  to  a  height  five  times  that  of 
the  fall  in  the  drive  pipe ;  this  is  a  rough  rule  only,  for  the 
length  of  the  discharge  pipe  is  one  of  the  controlling  factors 
as  well  as  its  vertical  rise. 

The  Rife  hydraulic  engine  is  a  water  ram  on  a  large  scale, 
two  or  more  being  connected  to  the  same  discharge  pipe  so 
that  the  flow  in  it  is  continuous.*  Three  of  these  engines  are 
said  to  raise  864  ooo  gallons  of  water  per  day  to  an  elevation 
of  150  feet,  the  fall  in  the  drive  pipe  being  30  feet.  The 
diameter  of  the  drive  pipe  is  8  inches  and  that  of  the  dis- 
charge pipe  is  4  inches;  the  waste  valve  weighs  50  pounds 
and  it  is  provided  with  an  adjusting  lever  in  order  that  its 
effective  weight  may  be  regulated  so  as  to  cause  the  maxi- 
mum discharge  to  be  delivered. 

*  Engineering  News,  1896,  vol.  36,  p.  429. 


ART.  190  OTHER  KINDS  OF  PUMPS  521 

Prob.  189a.  The  supply  of  air  in  the  air  chamber  D  is  main- 
tained by  having  a  small  hole  in  the  pipe  near  B.  Explain  the 
phenomena  and  the  reasons  thereof. 

Prob.  1896.  A  hydraulic  ram  raises  32^  pounds  of  water  in 
5  minutes  through  a  discharge  pipe  60  feet  long.  The  drive  pipe 
is  38  feet  long  and  the  amount  of  water  wasted  in  5  minutes  is 
41  \  pounds.  The  fall  of  the  drive  pipe  is  12  feet  and  the  vertical 
rise  of  the  discharge  pipe  above  the  ram  is  24  feet.  Compute 
the  efficiency  of  the  ram. 

ART.  190.     OTHER  KINDS  OF  PUMPS 

The  lift  and  force  pumps  described  in  Arts.  184  and  185 
are  called  displacement  pumps,  because  the  volume  of  water 
lifted  in  one  stroke  is  that  displaced  by  the  piston  or  plunger. 
If  there  be  no  leakage  past  the  piston  packing,  and  if  no  air 
is  mingled  with  the  water,  the  discharge  in  a  given  time 
may  be  very  accurately  determined  by  counting  the  number 
of  strokes  and  multiplying  this  number  by  the  displacement 
in  one  stroke.  On  account  of  the  reciprocating  motion  of 
the  piston  these  forms  are  often  called  reciprocating  pumps. 
There  is  always  a  loss  of  energy  due  to  putting  the  piston 
into  motion  at  the  beginning  of  each,  stroke,  and  to  avoid 
this  many  forms  of  rotary  pumps  have  been  devised,  yet 
notwithstanding  this  loss  the  plunger  force  pump  is  probably 
the  most  efficient  and  economical  of  all  kinds. 

A  rotary  or  impeller  pump  is  one  in  which  the  moving 
parts  have  a  circular  motion  only,  and  the  centrifugal  pump 
described  in  Art.  188  is  of  this  kind.  Numerous  other 
rotary  pumps  have  been  invented  but  none  is  widely  used 
except  the  centrifugal  one.  Fig.  190a  shows  one  where 
the  moving  parts  consist  of  two  wheels  which  are  rotated 
in  opposite  directions  as  indicated  by  the  arrows;  this 
motion  produces  a  partial  vacuum  whereby  the  water  rises 
in  the  suction  pipe  D,  and  is  then  carried  between  the  teeth 
and  the  case  and  forced  up  the  discharge  pipe  E.  Fig.  1906 
shows  a  form  where  the  moving  parts  are  two  lobes  or 


522 


PUMPS  AND  PUMPING 


CHAP.  XVI 


impellers  so  shaped  that  they  are  always  in  contact  with 
each  other  and  each  in  contact  with  the  enclosing  case.  In 
the  left-hand  diagram  the  water  rising  in  the  'pipe  D  is 


FIG.  190a 


FIG.  1906 


flowing  toward  the  right,  but  a  moment  later  the  lobe  B 
has  assumed  the  position  shown  in  the  right-hand  diagram 
and  the  water  is  imprisoned  between  the  lobe  and  the  case. 
An  instant  later  the  two  lobes  are  forcing  this  water  up 
the  pipe  E  while  the  water  coming  in  at  D  is  flowing  to  the 
left.  The  greatest  objection  to  these  pumps  is  that  it  is 
difficult  to  maintain  close  contact  between  the  case  and 
the  lobes  or  wheels,  owing  to  wear,  so  that  after  being  in 
use  for  some  time  there  is  much  back  leakage  of  water  and 
the  capacity  and  efficiency  of  the  pump  are  diminished.  The 
only  apparent  advantage  of  the  rotary  pump  is  that  it 
has  no  valves.  Five  rotary  pumps  of  the  type  of  Fig.  1906 
were  installed  in  1902  at  a  pumping  station  near  Chicago, 
the  lobes  or  impellers  being  4  feet  long  and  the  distance 
between  their  centers  2.7  feet;  these  pumps  run  at  100 
revolutions  per  minute  and  each  has  a  capacity  of  6000  cubic 
feet  per  minute  under  the  total  lift  of  about  8  feet.* 

The  pumps  thus  far  described,  with  the  exception  of 
the  hydraulic  ram,  maybe  called  mechanical  pumps,  because 
they  act  under  energy  communicated  to  them  from  motors. 
All  mechanical  pumps  are  reversible,  that  is,  when  the 

*  Engineering  News,  1903,  vol.  49,  p.  172. 


ART.  190  OTHER  KINDS  OF  PUMPS  523 

water  moves  in  the  opposite  direction  under  a  pressure- 
head  they  become  hydraulic  motors.  The  reverse  of  the 
chain  and  bucket  pump  is  the  overshot  or  breast  wheel, 
that  of  the  suction  and  lift  pump  is  the  water-pressure 
engine,  and  that  of  the  centrifugal  pump  is  the  turbine. 
The  hydraulic  ram  does  not  operate  under  the  action  of  a 
motor,  and  it  does  not  appear  to  be  reversible. 

Pumps  which  have  no  moving  parts  and  which  operate 
through  the  action  of  air  suction  and  dynamic  pressure 
constitute  another  class  which  will  now  be  briefly  con- 
sidered. Here  belong  the  jet  or  injector  pumps  which  act 
largely  through  suction,  and  the  injector  pump  used  on 
locomotives.  The  latter  pro- 
duces a  vacuum  through  the 
flow  of  steam,  and  cannot  be  dis- 
cussed here,  as  it  involves  prin- 
ciples of  thermodynamics.  The 
fundamental  principle,  however, 
is  indicated  in  Fig.  190c,  which 
shows  the  jet  apparatus  in- 
vented by  James  Thomson  in  FlG-  190c 

1850.*  The  water  to  be  lifted  is  at  C,  and  it  rises  by  suction 
to  the  chamber  J?,  from  which  it  passes  through  the  discharge 
pipe  to  the  tank  D.  The  forces  of  suction  and  pressure  are 
produced  by  a  jet  of  water  issuing  from  a  nozzle  at  the 
mouth  of  the  discharge  pipe,  the  nozzle  being  at  the  end 
of  a  pipe  A  B  through  which  water  is  brought  from  a  reser- 
voir ;  or  the  water  delivered  from  the  nozzle  may  come  from 
a  hydrant  or  from  a  force  pump.  Let  H  be  the  effective 
head  of  the  jet  as  it  issues  from  the  nozzle,  h±  the  suction 
lift,  and  h2  the  lift  above  the  tip  of  the  nozzle  ;  let  q  be  the 
discharge  through  the  nozzle  and  ql  that  through  the  suction 
pipe.  Then,  neglecting  frictional  resistances, 


.rJ 


*  Report  of  British  Association,  1852,  p.  130. 


524  PUMPS  AND  PUMPING  CHAP,  xvi 

and  the  efficiency  of  the  apparatus  is 
e=q1(hl+h2)/qH 

It  is  found  by  experiments  that  the  efficiency  of  this  jet 
pump  is  very  low,  usually  not  exceeding  20  percent,  the 
highest  efficiencies  being  for  low  ratios  of  ht  -f  h2  to  H .  This 
form  of  pump  has,  however,  been  found  very  convenient 
in  keeping  coffer  dams  and  sewer  trenches  free  from  water, 
as  it  requires  little  or  no  attention  and  has  no  moving  parts 
to  get  out  of  order. 

Another  class  of  pumps  uses  the  pressure  of  air  or  of 
steam  in  order  to  elevate  water.  The  idea  of  these  pumps 
is  old,  yet  it  was  not  until  1875  that  the  steam  pulsometer 
was  perfected  by  Hall,  while  the  air-lift  pump  of  Frizell 
dates  from  1880.  The  air-lift  pump  is  now  extensively 
used  for  raising  water  from  deep  wells,  the  compressed  air 
being  forced  down  a  vertical  pipe  in  the  well  tube  and 
issuing  from  its  lower  end.  As  it  issues,  bubbles  are  formed 
in  the  entire  column  of  water  in  the  well  tube,  and  being 
lighter  than  a  column  of  common  water,  it  rises  to  a  greater 
height  under  the  atmospheric  pressure,  assisted  by  the 
upward  impulse  of  the  bubbles  to  a  slight  extent.  In  this 
manner  water  having  a  natural  level  50  feet  or  more  below 
the  surface  of  the  ground  may  be  caused  to  rise  above  that 
surface.  It  has  been  found  in  practice  that  for  lifts  of  15 
to  50  feet  from  2  to  3  cubic  feet  of  air  are  necessary  for 
each  cubic  foot  of  water  that  is  elevated.  The  efficiency 
of  this  form  of  pump  is  low,  rarely  reaching  30  percent, 
although  a  maximum  of  50  percent  has  been  claimed.* 

Among  the  many  forms  of  pumps  operating  under  the 
pressure  of  compressed  air  only  the  ejector  pump  used  in 
the  Shone  system  of  sewerage  can  here  be  mentioned.  The 
sewage  from  a  number  of  houses  flows  to  a  closed  basin, 
called  an  injector,  in  which  it  continues  to  accumulate  until 
a  valve  is  opened  by  a  float.  The  opening  of  this  valve 
allows  compressed  air  to  enter  and  this  drives  out  the  sew- 

*  Journal  of  Association  of  Engineering  Societies,  1900,  vol.  25,  p.  173. 


ART.  191  PUMPING  THROUGH  PIPES  525 

age  through  a  discharge  pipe  to  the  place  where  it  is  desired 
to  deliver  it.  'In  the  installation  of  this  system  of  sewerage 
at  the  World's  Fair  of  1893  in  Chicago,  there  were  26 
ejectors  which  lifted  the  sewage  67  feet,  the  total  pressure- 
head  being  about  108  feet.  Vacuum  methods  of  moving 
sewage  have  also  been  used  in  Europe,  but  these  cannot 
compete  in  efficiency  with  those  using  compressed  air. 

Prob.  190.  For  Fig.  190^  let  the  diameter  of  the  nozzle  be 
i  inch  and  that  of  the  discharge  pipe  4  inches.  Let  H  be  64 
feet,  h^  be  1 8  feet,  h.2  be  3  feet,  and  the  discharge  from  the  nozzle 
be  0.25  cubic  feet  per  second.  Compute  the  greatest  quantity 
of  water  that  can  be  lifted  per  second  through  the  suction  pipe, 
and  the  efficiency  of  the  apparatus  when  doing  this  work. 

ART.  191.     PUMPING  THROUGH  PIPES 

When  water  is  pumped  through  a  pipe  from  a  lower  to  a 
higher  level,  the  power  of  the  pump  must  be  sufficient  not 
only  to  raise  the  required  amount  in  a  given  time,  but  also 
to  overcome  the  various  resistances  to  flow.  The  head  due 
to  the  resistances  is  thus  a  direct  source  of  loss,  and  it  is 
desirable  that  the  pipe  be  so  arranged  as  to  render  this  as 
small  as  possible.  The  length  of  the  pipe  is  always  much 
greater  than  the  vertical  lift,  so  that  the  losses  of  head  in 
friction  are  materially  higher  than  those  indicated  by  the 
discussion  of  Art.  186,  where  vertical  pipes  were  alone  con- 
sidered. 

Let  w  be  the  weight  of  a  cubic  foot  of  water  and  q  the 
quantity  raised  per  second  through  the  height  h,  which,  for 
example,  may  be  the  dif- 
ference in  level  between 
a  canal  C  and  a  reservoir 
R,  as  in  Fig.  19  la.     The 
useful  work  done  by  the 
pump  in  each  second  is 
wqh.     Let  h'  be  the  head 
lost  in  entering  the  pipe  at  the  canal,  h"  that  lost  in  friction 


526  PUMPS  AND  PUMPING  CHAP,  xvi 

in  the  pipe,  and  h'"  all  other  losses  of  head,  such  as  those 
caused  by  curves,  valves,  and  by  resistances  in  passing 
through  the  pump  cylinders.  Then  the  total  work  per- 
formed by  the  pump  per  second  is 


k  =  wqh  +  wq(h'+h"+h"')  (191), 

Inserting  the  values  of  the  lost  heads  from  Arts.  85-88,  this 
expression  takes  the  form 


in  which  v  is  the  velocity  in  the  pipe,  /  its  length,  and  d  its 
diameter.  In  order,  therefore,  that  the  losses  of  work  may 
be  as  small  as  possible,  the  velocity  of  flow  through  the  pipe 
should  be  low;  and  this  is  to  be  effected  by  making  the 
diameter  of  the  pipe  large.  The  size  of  the  pipe  is  here  re- 
garded as  uniform  from  the  canal  to  the  reservoir;  in  prac- 
tice the  suction  pipe  is  usually  larger  in  diameter  than  the 
discharge  pipe,  in  order  that  the  suction  valves  may  receive 
an  ample  supply  of  water. 

For  example,  let  it  be  required  to  determine  the  horse- 
power of  a  pump  to  raise  i  200  ooo  gallons  per  day  through 
a  height  of  230  feet  when  the  diameter  of  the  pipe  is  6  inches. 
and  its  length  1400  feet.  The  discharge  per  second  is 

i  200  ooo 

q  =  -  —  5—  —7  —  =  1.86  cubic  feet, 

7.481 


and  the  velocity  in  the  pipe  is 

1.86 

oT2  =9'47 
The  probable  head  lost  in  entering  the  pipe  is,  by  Art.  85, 

v2 
hf  =0.5—  =0.5  Xi-39  =0.7  feet. 

When  the  pipe  is  new  and  clean  the  friction  factor  /  is  about 


ART.  191  PUMPING  THROUGH  PIPES  527 

0.020,  as  shown  by  Table  33;  then  the  loss  of  head  in  fric- 
tion in  the  pipe  is,  by  Art.  86, 


/*"=  0.020  X  X  1.39  =77.8  feet. 

The  other  losses  of  head  depend  upon  the  details  of  the  pump 
cylinder  and  the  valves;  if  these  be  such  that  ^,  =  4,  then 

#"=4X1.39  =5.6  feet. 
The  total  losses  of  head  hence  are 

#  +  #"  +  #"=84.1  feet. 

The  work  to  be  performed  per  second  by  the  pump  now  is 
£=62.5  XL  86(230  +  84.1)  =36  510  foot-pounds, 

and  the  horse-power  to  be  expended  is  36  51  0/550=66.  4. 
If  there  were  no  losses  in  friction  and  other  resistances  the 
work  to  be  done  would  be  simply 

k  =62.5  Xi.86  X23O  =  26  740  foot-pounds, 

and  the  corresponding  horse-power  would  be  26  740/550  = 
48.6.  Hence  17.8  horse-power  is  wasted  in  injurious  resist- 
ances, or  the  efficiency  of  the  plant  is  only  73  percent. 

For  the  same  data  let  the  6  -inch  pipe  be  replaced  by  one 
14  inches  in  diameter.  Then,  proceeding  as  before,  the 
velocity  of  flow  is  found  to  be  1.74  feet  per  second,  the  head 
lost  at  entrance  0.03  feet,  the  head  lost  in  friction  1.13  feet, 
and  that  lost  in  other  ways  0.19  feet.  The  total  losses  of 
head  are  thus  only  1.35  feet,  as  against  84.1  feet  for  the 
smaller  pipe,  and  the  horse-power  required  is  48.9,  which  is 
but  little  greater  than  the  theoretic  power.  The  great  ad- 
vantage of  the  larger  pipe  is  thus  apparent,  and  by  increasing 
its  size  to  1  8  inches  the  losses  of  head  may  be  reduced  so  low 
as  to  be  scarcely  appreciable  in  comparison  with  the  useful 
head  of  230  feet. 


528  PUMPS  AND  PUMPING  CHAP,  xvi 

A  pump  is  often  used  to  force  water  directly  through  the 
mains  of  a  water-supply  system  under  a  designated  pressure. 

The  work  of  the  pump  in  this 
case  consists  of  that  required 
to  maintain  the  pressure  and 
that  required  to  overcome  the 
frictional  resistances.  Let  ht 
be  the  pressure-head  to  be 
maintained  at  the  end  of  the 
main,  and  z  the  height  of  the 
FIG.  1916  main  above  the  level  of  the 

river  from  which  the  water  is  pumped;  then  h^  +  z  is  the 
head  H,  which  corresponds  to  the  useful  work  of  the  pump, 
and,  as  before, 


To  reduce  the  injurious  heads  to  the  smallest  limits  the 
mains  should  be  large  in  order  that  the  velocity  of  flow  may 
be  small.  In  Fig.  1916  is  shown  a  symbolic  representation 
of  the  case  of  pumping  into  a  main,  P  being  the  pump,  C 
the  source  of  supply,  and  DM  the  pressure-head  which  is 
maintained  upon  the  end  of  the  pipe  during  the  flow.  At 
the  pump  the  pressure-head  is  AP,  so  that  AD  represents 
the  hydraulic  gradient  for  the  pipe  from  P  to  M.  The  total 
work  of  the  pump  may  then  be  regarded  as  expended  in 
lifting  the  water  from  C  to  A,  and  this  consists  of  three  parts 
corresponding  to  the  heads  CM  or  z,  MD  or  hlt  and  A  B  or 
h'  +  h"  +  h'",  the  first  overcoming  the  force  of  gravity,  the  sec- 
ond maintaining  the  discharge  under  the  required  pressure, 
while  the  last  is  transformed  into  heat  in  overcoming  fric- 
tion and  other  resistances.  In  this  direct  method  of  water 
supply  a  standpipe,  AP,  is  often  erected  near  the  pump,  in 
which  the  water  rises  to  a  height  corresponding  to  the  re- 
quired pressure,  and  which  furnishes  a  supply  when  a  tem- 
porary stoppage  of  the  pumping  engine  occurs.  This  stand- 
pipe  also  relieves  the  pump  to  some  extent  from  the  shock 
of  water  hammer  (Art.  148). 


ART.  192  PUMPING   THROUGH   HOSE  529 

Prob.  191.  Compute  the  horse-power  of  a  pump  for  the  fol- 
lowing data,  neglecting  all  resistances  except  those  due  to  pipe 
friction:  #  =  1.5  cubic  feet  per  second,  which  is  distributed  uni- 
formly over  a  length  Zt  =  3000  feet  (Art.  99),  the  remaining  length 
of  the  pipe  being  4290  feet;  d=io  inches,  ^  =  75.8  feet,  and 
2=10.6  feet. 

ART.  192.      PUMPING  THROUGH  HOSE 

In  Art.  102  the  flow  of  water  through  fire  hose  was 
briefly  treated  and  the  friction  factors  given  for  different 
kinds  of  hose  linings.  It  was  shown  that  the  loss  of  head 
in  a  long  hose  line  becomes  so  great,  even  tinder  moderate 
velocities,  as  to  consume  a  large  proportion  of  the  pressure 
exerted  by  the  hydrant  or  steamer.  As  another  example, 
let  the  pressure  in  the  pump  of  the  fire  engine  be  1 2  2  pounds 
per  square  inch,  corresponding  to  a  head  of  281  feet,  and  let 
it  be  required  to  find  the  pressure-head  in  2^-inch  rough 
rubber-lined  cotton  hose  at  1000  feet  distance,  when  a  noz- 
zle is  used  which  discharges  153  gallons  per  minute,  the  hose 
being  laid  horizontal.  The  discharge  is  0.341  cubic  feet  pet- 
second,  which  gives  a  velocity  of  10.0  feet  per  second  in  the 
hose.  Hence  by  (86)  the  loss  of  head  in  friction  is  231  feet, 
so  that  the  pressure-head  at  the  nozzle,  entrance  is  only  50 
feet,  which  corresponds  to  about  22  pounds  per  square  inch. 
The  remedy  for  this  great  reduction  of  pressure  is  to  employ 
a  smaller  nozzle,  thus  decreasing  the  discharge  and  the  ve- 
locity in  the  hose ;  but  if  both  head  and  discharge  are  desired 
they  may  be  obtained  either  by  an  increase  of  pressure  at 
the  steamer  or  by  the  use  of  a  larger  hose. 

Another  method  of  securing  both  high  velocity-head  and 
quantity  of  water  is  by  the  use  of  siamesed  hose  lines,  and 
this  is  generally  used  when  large  fires  occur.  This  method 
consists  in  having  several  lines  of  hose,  generally  four,  lead 
from  the  steamer  to  a  so-called  Siamese  connection,  from 
which  a  short  single  line  of  hose  leads  to  the  nozzle.  In  Fig. 
192  the  pump  or  fire  steamer  is  represented  by  A,  the 


530  PUMPS  AND  PUMPING  CHAP,  xvi 

Siamese  joint  by  B,  the  nozzle  entrance  by  C,  and  the  nozzle 
tip  by  D.  From  A  let  n  lines  of  hose,  each  having  the 
length  /!  and  the  diameter  dlt  lead  to  B ;  and  from  B  let  there 
be  a  single  line  of  length  /2  and  diameter  d2  leading  to  the 
nozzle  which  has  the  diameter  D.  The  hydraulic  gradient 
(Art.  95)  is  shown  by  abcD,  the  pressure-heads  at  A,  B,  C 


£ 

*i         """*;.  i  \ 

i 


-^B  G    D 

FIG.  192 

being  represented  by  Aa,  Bb,  Cc.  Let  h  be  the  pressure- 
head  on  the  nozzle  tip  or  the  difference  of  the  elevations  of 
the  points  a  and  D.  It  is  required  to  deduce  a  formula 
for  the  velocity  at  the  nozzle  tip  and  to  determine  the  pres- 
sure-heads at  B  and  C. 

This  case  is  one  of  diversions,  already  treated  in  Art. 
100,  and  the  sam'e  principles  may  be  applied  to  its  solution. 
Neglecting  losses  in  entrance,  in  curvature,  and  in  the 
Siamese  joint,  the  total  head  h  is  expended  in  friction  in  the 
hose  lines  and  in  the  nozzle,  or 

ax  2g         d2  2g      cl   2g 

in  which  vt  and  v2  are  the,  velocities  in  the  lines  ll  and  12>  and 
V  is  that  from  the  nozzle,  while  c1  is  the  coefficient  of  ve- 
locity of  the  nozzle  (Art.  80).  The  first  term  of  the  second 
member  is  the  head  lost  between  A  and  B,  and  the  alge- 
braic expression  for  this  is  independent  of  the  number  of 
hose  lines  between  those  points;  the  velocity  v1  in  these 
hose  lines  depends,  indeed,  upon  their  number,  but  the 
hydraulic  gradient  ab  is  the  same  for  each  and  all  of  them. 
The  law  of  continuity  of  flow  (Art.  32)  gives,  however, 

nd12vi=d22v2=D2V 
and,  taking  from  these  the  values  of  vl  and  v2  in  terms  of  V 


.  192  PUMPING  THROUGH  HOSE  531 


and  inserting  them  in  the  expression  for  h,  there  results 


d,d 


(192) 


from  which  the  velocity  V  and  the  velocity-head  V2/2g 
can  be  computed,  while  the  discharge  is  given  by  q  =  \KD*V. 
The  pressure-head  h2  at  the  nozzle  entrance  and  the  pressure- 
head  h^  at  the  Siamese  joint  may  then  be  found  from 


and,  as  a  check,  the  latter  should  equal  h  minus  the  drop  of 
the  hydraulic  gradient  between  a  and  b. 

This  discussion  shows  that,  by  increasing  the  number  n, 
the  loss  of  head  between  A  and  B  may  be  made  very  small, 
the  effect  being  practically  the  same  as  that  of  moving  the 
steamer  to  B  and  using  but  a  single  hose  line  /2.  As  a  nu- 
merical example,  let  h  =  230.4  feet,  /x  =  500  feet,  12  =60  feet, 
^i  =d2  =  2.5  inches,  D  =  i  inch,  and  c1  =0.975.  Then,  taking 
/  as  0.03,  the  computed  results  for  different  values  of  n  are 
as  follows,  V  being  in  feet  per  second,  V2/2g  in  feet,  and  q 

n=    12346  oo 

V=68.9       92.2       99.8       10.3       105  107 

t/2/2g  =  73-7     i32         !55         165           173  180 

2=169      226        244        252           258  263 

in  gallons  per  minute.  It  is  seen  that  for  four  lines  the  veloc- 
ity-head is  more  than  double  that  for  a  single  line  and  that 
the  discharge  is  50  percent  greater.  With  more  than  four 
lines  the  velocity-head  and  discharge  increase  slowly,  and 
for  n  =  oo  they  are  practically  the  same  as  for  n  =  10.  The 
number  of  hose  lines  generally  used  is  four,  since  the  slight 
advantage  of  more  lines  is  not  sufficient  to  warrant  their  use. 

Many  other  interesting  problems  relating  to  hose  lines 
may  be  solved  by  using  the  same  principles.  If  there  be 
four  lines  of  hose  between  the  pump  and  the  Siamese  joint, 


532  PUMPS   AND    DUMPING  CHAP.  XVI 

three  having  the  diameter  d1  and  one  having  the  diameter 
d,  it  can  be  shown  that  the  formula  (192)  applies,  provided 
n  be  replaced  by  3  +  (d/d$*  For  instance,  if  d  be  3  inches 
and  d±  be  2\  inches,  this  makes  n2  about  19.  In  deducing 
this  expression  for  n  it  is  assumed  that  the  friction  factors 
are  the  same  for  both  sizes  of  hose,  although  in  strictness 
the  smaller  hose  has  the  higher  value  of  /. 

Another  case  is  where  two  of  the  hose  lines  between  A 
and  B  have  the  diameter  dl  and  the  length  /t,  while  the  two 
other  lines  are  of  the  length  l  +  ly  the  length  /  having  the 
diameter  d  and  the  length  Z3  the  diameter  d3.  Here  the 
principles  regarding  compound  pipes  (Art.  96)  are  also  to 
be  regarded,  and  it  will  be  found  that  formula  (192)  applies 
likewise  to  this  case,  if  n  be  computed  from 


in  which  e  represents  f(l/d),  while  e1  and  e3  represent 
and  fa(la/dB)  respectively.  For  instance,  if  /t  =  100,  13  =  ioo> 
and  /  =  5o  feet,  while  d1=da  =  2$  inches  and  d  =  $  inches, 
then  the  value  of  n2  is  about  21,  so  that  this  arrangement 
is  more  effective  than  that  of  the  preceding  paragraph. 

In  the  deduction  of  the  above  formulas  losses  of  head 
at  entrance  and  in  the  Siamese  joint  have  not  been  regarded, 
and  it  is  unnecessary  to  consider  these  when  the  hose  lines 
are  long.  For  lines  less  than  100  feet  in  length  the  losses 
of  head  at  entrance  may  be  taken  into  account  by  adding 
the  term  o.^(D/d1)2/n2  to  the  denominator  of  (192).  The 
loss  of  head  due  to  the  Siamese  joint  may,  in  the  absence 
of  experimental  data,  be  approximately  accounted  for  by 
adding  about  0.02  to  that  denominator,  thus  considering 
its  influence  about  one-half  that  of  the  nozzle.  In  a  case 
like  that  of  the  last  paragraph,  where  the  length  /  in  two 
of  the  hose  lines  is  nearest  the  pumps,  the  values  of  e  and  et 
may  be  increased  by  0.5  in  order  to  introduce  the  influence 
of  the  entrance  heads.  Errors  of  5  percent  or  more  are 


ART.  192  PUMPING   THROUGH    HOSE  533 

liable  to  occur  in  computations  on  pumping  through  short 
hose  lines. 

Prob.  192a.  Three  hose  lines  run  from  a  pump  to  a  Siamese 
connection,  each  being  500  feet  long  and  2^  inches  in  diameter, 
and  from  the  Siamese  one  line  50  feet  long  and  2\  inches  in  diam- 
eter leads  to  a  i -J-inch  nozzle  having  a  velocity  coefficient  of  0.96. 
When  the  pressure  at  the  pump  is  100  pounds  per  square  inch, 
what  is  the  discharge  from  the  nozzle  and  the  velocity-head  of 
the  jet?  What  friction  heads  are  lost  in  the  hose  and  nozzle? 

Prob.  1926.  In  a  fire-engine  test  made  in  1903,  the  lengths  lt 
and  13  were  50  feet,  the  length  /  was  12  feet,  and  /2  was  zero,  as 
the  nozzle  was  attached  directly  to  the  Siamese  joint.  The 
diameter  dl  was  3  inches,  while  d  and  d3  were  2  J  inches,  and  D 
was  2  inches.  The  pressure  gage  on  the  steamer  read  90,  while 
one  on  the  siamese  joint  read  63  pounds  per  square  inch.  Com- 
pute the  pressure-head  at  the  siamese  joint. 

Prob.  192c.  What  is  the  efficiency  of  a  bucket  pump  which 
lifts  2000  liters  of  water  per  minute  through  a  height  of  3.5 
meters  with  an  expenditure  of  2.5  metric  horse-powers? 

Prob.  192J.  When  the  height  of  the  mercury  barometer  is 
760  millimeters,  water  at  a  temperature  of  o°  centigrade  is 
raised  by  suction  in  a  perfect  vacuum  to  a  height  of  10.33  meters 
(Art.  184).  Under  the  same  atmospheric  pressure,  how  high 
can  it  be  raised  when  the  temperature  is  32°  centigrade? 

Prob.  192^.  What  metric  horse-power  is  required  to  raise 
4  ooo  ooo  liters  per  day  through  a  height  of  75  meters  when  the 
diameter  of  the  pipe  is  20  centimeters  and  its  length  500  meters? 

Prob.  192/.  The  calorie  is  the  metric  thermal  unit,  this  being 
the  energy  required  to  raise  one  kilogram  of  water  one  degree 
centigrade  when  the  temperature  of  the  water  is  near  that  of 
maximum  density.  How  many  calories  are  equivalent  to 
i  ooo  ooo  English  thermal  units? 


534  APPENDIX  ART.  193 


APPENDIX 

ART.  193.     HYDRAULIC-ELECTRIC  ANALOGIES 

It  is  well  known  that  there  are  certain  analogies  between 
the  flow  of  water  in  pipes  and  that  of  the  electric  current 
in  wires,  and  some  of  these  will  here  be  briefly  explained 
from  a  hydraulic  point  of  view.  The  electric  analog  of  a 
water  pump  is  the  dynamo,  both  being  driven  by  mechan- 
ical power  and  both  transforming  it  into  other  forms  of 
energy.  The  analog  of  a  water  wheel  is  the  electric  motor, 
each  of  which  delivers  mechanical  power  by  virtue  of  the 
energy  transmitted  to  it  through  the  water  pipe  or  electric 
wire.  While  the  water  is  flowing  from  the  pump  to  the 
wheel  much  of  its  energy  is  lost  in  overcoming  frictional 
resistances,  whereby  heat  is  produced ;  while  the  electricity 
is  flowing  from  the  dynamo  to  the  electric  motor  some  of 
its  energy  is  lost  in  overcoming  molecular  resistances, 
whereby  heat  is  produced.  The  steady  flow  of  water  cor- 
responds to  the  continuous  flow  of  electricity  in  one  direc- 
tion, or  to  the  direct  current,  and  the  following  discussion 
compares  hydraulic  phenomena  with  those  of  the  direct 
electric  current.  The  phenomena  of  the  alternating  cur- 
rent have  also  certain  hydraulic  analogies,  but  these  will 
not  be  discussed  here. 

Let  q  represent  electric  current,  R  the  electric  resistance 
of  a  wire  of  length  Z,  cross-section  a,  and  diameter  d,  and  p 
the  electromotive  force  under  which  the  current  is  pushed 
through  the  wire.  Then  Ohm's  law  gives,  if  s  be  the  specific 
resistance  of  the  material  of  the  wire, 

(193), 


ART.  193  HYDRAULIC-ELECTRIC   ANALOGIES  535 

in  which  A  is  a  constant  depending  only  on  the  material  of 
the  wire.  This  equation  shows  that  the  electric  pressure 
p  varies  directly  with  the  length  of  the  wire,  inversely  as 
the  square  of  its  diameter,  and  directly  as  the  current.  By 
increasing  the  length  of  the  wire  or  by  decreasing  its  diam- 
eter, the  electromotive  force  required  to  maintain  a  given 
electric  current  is  increased.  Similarly  in  a  water  pipe  the 
friction-head  required  to  maintain  a  given  discharge  in- 
creases directly  as  the  length  of  the  pipe,  and  is  greater  for 
a  small  pipe  than  for  a  large  one  (Art.  86). 

In  Art.  100  it  was  pointed  out  that  the  distribution  of 
water  flow  among  several  diversions  of  a  pipe  follows  laws 
analogous  to  those  of  the  electric  current.  It  was  there 
shown  that  the  discharge  q  divides  between  the  diversions 
inversely  as  their  resistances,  provided  (fl/d5)^  be  taken 
as  the  measure  of  resistance.  In  electric  flow  the  direct 
current  is  the  analog  of  the  discharge  in  the  water  pipe,  but 
Ohm's  law  shows  that  the  resistance  is  the  simpler  quantity 
fl/d2.  The  hydraulic  analog  of  electromotive  force  is  often 
taken  to  be  the  lost  friction -head  or  its  corresponding  unit 
pressure,  and  this  will  be  followed  here.  The  loss  in  water 
pressure  is  represented  by  the  hydraulic  gradient  (Art.  95), 
and  the  loss  in  electric  pressure  is  often  represented  in  a  sim- 
ilar way,  the  gradient  being  a  straight  line  in  both  cases. 

In  order  to  make  an  algebraic  comparison  of  the  two 
phenomena,  take  the  expression  for  friction-head  in  (86)  and 
replace  h"  by  p/w,  where  p  is  the  loss  of  unit  pressure  in 
the  length  /,  and  w  is  the  weight  of  a  cubic  unit  of  water ; 
also  replace  v  by  q/a,  and  a  by  \xd2.  Then  (86)  becomes 

P-^^-BZ?  (193), 

in  which  the  constant  B  depends  upon  the  roughness  of  the 
surface  and  the  force  of  gravity.  Accordingly  the  lost 
pressure  varies  directly  as  the  length  of  the  pipe,  inversely 
as  the  fifth  power  of  its  diameter,  and  directly  as  the  square 
of  the  discharge. 


536  APPENDIX  ART.  193 

Thus,  in  the  case  of  a  single  water  pipe  or  electric  wire, 
for  electric  flow  p=A^q 

for  hydraulic  flow       p=B-^q2 

If  each  of  these  flows  be  divided  among  n  diversions,  as  in 
Fig.  192,  the  expressions  for  the  pressure  become 

Al 
for  electric  flow  P=~~T2CL 

7?Z 
for  hydraulic  flow        p  =-^Tb(f 

so  that  the  drop  of  the  gradient  is  far  more  rapid  in  the 
latter  case ;  thus,  if  n  be  3  the  electromotive  force  for  three 
wires  is  one-third  of  that  for  a  single  wire,  but  the  hydraulic 
pressure  for  three  pipes  is  one-ninth  of  that  for  a  single  pipe. 

The  conclusion  to  be  derived  from  this  comparison  is 
that  the  analogies  between  hydraulic  and  electric  flow  are 
rough  ones  and  cannot  embrace  all  the  quantities  involved. 
The  only  perfect  analogy  is  that  p  varies  directly  as  / ;  the 
analogy  between  hydraulic  discharge  and  electric  current  is 
perfect  only  as  regards  its  distribution  between  branches  or 
diversions;  the  analogy  between  hydraulic  and  electric  re- 
sistance is  an  imperfect  one  that  is  liable  to  lead  to  con- 
fusion. Although  a  decrease  in  size  of  the  pipe  or  wire 
causes  an  increase  in  resistance,  the  law  of  increase  is  quite 
different  in  the  two  cases.  If  hydraulic  resistance  be  de- 
fined as  in  Art.  100,  then  the  lost  pressure  p  is  not  pro- 
portional to  resistance,  but  to  its  square  root,  while  the  lost 
electric  pressure  p  varies  directly  as  electric  resistance. 

Thus  far  the  common  motion  of  water  in  pipes  has  been 
considered,  namely,  flow  at  such  velocities  as  are  usual  in 
engineering  practice.  In  Art.  103,  however,  it  was  noted 
that  the  laws  governing  the  flow  of  water  at  low  velocities 
are  very  different,  the  lost  head  varying  directly  as  the  ve- 
locity and  inversely  as  the  area  of  the  cross-sections.  This 


ART.  193  HYDRAULIC-ELECTRIC   ANALOGIES  537 

kind  of  flow  may  be  called  viscous,  implying  that  the  resist- 
ances are  those  of  sliding  friction  only  and  that  no  losses 
occur  in  impact.  For  this  case  of  viscous  hydraulic  flow  let 
h  be  replaced  by  p/w  and  v  by  q/a  ;  then 

I         ^  I 


which  shows  that  the  lost  pressure  is  proportional  to  the 
discharge  as  in  Ohm's  law.  Further  in  this  viscous  flow  the 
product  of  the  lost  unit  pressure  p  and  the  discharge  q  is 
energy  per  second  ;  in  electric  flow  also  the  product  of  the 
lost  voltage  p  and  the  current  q  is  energy  per  second.  The 
formal  algebraic  expressions  for  the  two  cases  agree,  except 
in  regard  to  the  resistance,  which  varies  inversely  as  the 
area  of  the  wire  in  electric  flow  and  inversely  as  the  square 
of  the  area  of  the  pipe  in  viscous  hydraulic  flow.  Thus  this 
analogy  breaks  down,  as  all  analogies  correlating  electric 
with  mechanical  phenomena  are  found  to  do  sooner  or  later.* 
Hydraulic  flow  can  be  directly  observed  by  the  senses; 
electric  flow,  whether  it  be  in  the  wire  or  in  the  ether  out- 
side of  the  wire,  can  be  only  indirectly  observed  ;  yet  in  both 
cases  energy  is  transmitted  or  transformed  into  heat.  Elec- 
tric phenomena  are  undoubtedly  manifestations  of  matter, 
ether,  and  motion  ;  and  electricity,  whatever  its  real  nature 
may  be,  is  governed  by  the  same  laws  of  energy  that  pervade 
all  branches  of  hydraulics. 

Prob.  193a.  A  copper  wire  having  a  specific  resistance  of 
0.0000016  ohms  is  12  kilometers  long  and  one  centimeter  in 
diameter.  Compute  the  loss  in  voltage  required  to  maintain 
a  direct  current  of  150  amperes. 

Prob.  1936.  Let  9  kilometers  of  the  above  line  be  copper 
wire,  and  3  kilometers  be  a  steel  rail  having  a  specific  resist- 
ance of  0.0000145  ohms  and  a  cross-section  of  8.5  square  inches. 
Compute  the  loss  in  voltage  to  maintain  the  current  of  150 
amperes.  If  the  pressure  at  the  beginning  of  the  line  be  2500 
volts  and  the  rail  section  be  at  the  middle  of  the  line,  draw 
the  electric  gradient. 

*  Heaviside,  Electromagnetic  Theory  (London,  1894),  vol.  i,  p.  232. 


538  APPENDIX  ART.  194 


ART.  194.     MISCELLANEOUS  PROBLEMS 

The  following  problems  introduce  subjects  that  have  not 
been  specifically  treated  in  the  preceding  pages.  Teach- 
ers who  wish  to  offer  prize  problems  to  their  classes  may 
perhaps  find  some  of  these  suitable  for  that  purpose. 

Prob.  194a.  A  wooden  water  tank  18  feet  in  diameter  and 
24  feet  high  is  to  be  hooped  with  iron  bands  which  may  be 
safely  spaced  6  inches  apart  at  the  middle  of  the  height.  How 
far  apart  should  they  be  spaced  at  the  bottom? 

Prob.  1946.  A  house  is  60  feet  lower  than  a  spring  A  and 
30  feet  higher  than  a  spring  B.  A  pipe  from  A  to  the  house 
runs  near  B.  Explain  a  method  by  which  the  water  from  B 
can  be  drawn  into  the  pipe  and  be  delivered  at  the  house. 

Prob.  194c.  A  river  having  a  width  of  300  feet  on  the  sur- 
face, a  cross-section  of  1800  square  feet,  a  hydraulic  radius  of 
5.3  feet,  and  a  slope  of  i  on  10  ooo,  discharges  10  400  cubic 
feet  per  second.  If  it  be  frozen  over  to  the  depth  of  one  foot, 
what  will  be  its  discharge? 

Prob.  194J.  From  a  pumping  station  water  is  forced  by 
direct  pressure  through  a  compound  pipe,  consisting  of  7500 
feet  of  i4-inch  pipe,  4100  feet  of  12 -inch  pipe,  and  780  feet  of 
8-inch  pipe,  to  a  6-inch  pipe  on  which  there  are  three  hydrants  A, 
B,  and  C.  A  is  133  feet  from  the  end  of  the  8-inch  pipe  and 
115  feet  above  the  gage  at  the  pumping  station ;  B  is  433  feet 
from  the  end  of  the  8-inch  pipe  and  135  feet  above  the  gage; 
C  is  733  feet  from  the  end  of  the  8-inch  pipe  and  125  feet  above 
the  gage.  To  each  of  these  hydrants  is  attached  50  feet  of 
2j-inch  rubber-lined  hoSe  with  a  i-inch  smooth  nozzle  at  the 
end.  When  the  gage  at  the  pumping  station  reads  175  pounds 
per  square  inch,  to  what  heights  will  the  three  streams  be 
thrown  from  the  three  nozzles? 

Prob.  194<?.  When  a  body  falls  vertically  in  water  its  ve- 
locity soon  becomes  constant.  For  a  smooth  sphere  an  approx- 
imate formula  for  this  velocity  is  v  =  \/2gd(s—  i),  in  which 
d  is  the  diameter  of  the  sphere  and  5  its  specific  gravity.  Com- 
pute the  velocity  v  for  a  sphere  having  a  diameter  of  o.ooi  feet 
and  a  specific  gravity  of  1.25. 


ART.  194  MISCELLANEOUS  PROBLEMS  539 

Prob.  194/.  The  velocity  with  which  water  flows  through 
a  sand  filter  bed  varies  directly  as  the  head  (Art.  103).  If  V 
be  the  velocity  in  meters  per  day,  d  the  effective  size  of  the 
sand  grains  in  millimeters,  h  the  head,  /  the  thickness  of  the 
sand  bed,  and  t  the  temperature  on  the  centigrade  scale, 

V=  1000(0.70  +  0.030(^/0^ 

is  the  formula  deduced  by  Hazen.*  When  2  =  32°.  4  centigrade, 
^  =  0.4  millimeters,  /  =  4  feet,  and  h  =  o.4  feet,  find  how  many 
million  gallons  per  day  will  pass  through  one  acre  of  filter  beds. 

Prob.  194g.  A  bent  U  tube  of  uniform  size  is  partly  filled 
with  water.  Let  the  water  in  one  leg  be  depressed  a  certain 
distance,  causing  that  in  the  other  to  rise  the  same  distance. 
When  the  depressing  force  is  removed  the  water  oscillates  up 
and  down  in  the  legs  of  the  tube,  the  times  of  oscillation  being 
isochronous.  If  /  be  the  entire  length  of  the  water  in  the  tube, 
show  that  the  time  of  one  oscillation  is  n\/l/2g.  If  the  legs 
are  inclined  to  the  horizontal  at  the  angles  d  and  <j>,  show  that 
the  time  of  one  oscillation  is  7rV7/g(sin#  +  sin</>). 

Prob.  194/z.  The  bottom  of  a  canal  has  the  width  26  and  it 
is  desired  to  shape  the  banks  so  that  the  hydraulic  radius  of 
the  cross-section  may  be  constant.  Show  that  the  equation  of 
the  curve  is 

y=r 


in  which  y  is  the  depth  of  the  water,  x  the  half  width  of  the 
water  surface,  and  r  the  constant  hydraulic  radius. 

Prob.  194t.  A  river  having  a  slope  of  i  on  2500  runs  due 
east.  A  line  drawn  due  north  at  a  point  A  on  the  river  strikes 
at  B,  5000  feet  from  A,  the  edge  of  a  large  swamp  which  it  is 
desired  to  drain.  The  level  of  the  water  in  this  swamp  is  0.5 
feet  below  the  river  surface  at  A  and  it  is  desired  to  lower  that 
level  1.5  feet  more.  For  this  purpose  a  ditch  is  to  be  dug  run- 
ning from  A  in  a  straight  line  on  a  uniform  slope  until  it  joins 
the  river  at  a  point  C  eastward  from  A  .  The  discharge  of  this 
ditch,  in  order  to  properly  drain  the  swamp,  will  be  25  cubic 
feet  per  second,  its  side  slopes  are  to  be  i  on  i,  the  mean  veloc- 
ity is  not  to  exceed  2.5  feet  per  second,  and  the  coefficient  c  in 
the  Chezy  formula  is  estimated  at  70.  Find  the  length  and 
width  of  the  most  economical  ditch. 

*  Report  Massachusetts  State  Board  of  Health,  1892,  p.  553. 


540  APPENDIX  ART.  195 

ART.  195.      ANSWERS  TO  PROBLEMS 

Below  will  be  found  answers  to  some  of  the  problems 
given  in  the  preceding  pages,  the  numbers  of  the  problems 
being  placed  in  parentheses.  In  general  it  is  not  a  good 
plan  for  a  student  to  solve  a  problem  in  order  to  obtain  a 
given  answer.  One  object  of  solving  problems  is,  of  course, 
to  obtain  correct  results,  but  the  correctness  of  those  results 
should  be  established  by  methods  of  verification  rather  than 
by  the  authority  of  a  given  answer.  It  is  more  profitable 
that  a  number  of  students  should  obtain  different  answers 
to  a  problem  and  engage  in  a  discussion  as  to  the  correctness 
of  their  solutions  than  that  all  discussion  should  be  stopped 
because  a  certain  answer  is  given  in  the  text.  However 
satisfactory  it  may  be  to  know  in  advance  the  result  of  the 
solution  of  an  exercise,  let  the  student  bear  in  mind  that  after 
commencement  day  answers  to  problems  will  not  be  given 
him,  while  the  collection  of  the  data  for  a  problem  will 
often  prove  as  difficult  as  its  solution.  The  remarks  in 
Art.  8  may  be  again  read  in  this  connection,  and  the  student 
is  urged  to  follow  the  advice  there  given.  An  answer  here 
given  should  in  no  event  be  consulted  until  the  student 
has  completed  the  solution  of  the  problem. 

(1)  See  Rankine's  Miscellaneous  Writings.  (2)  46.5  horse- 
powers. (3)  See  Tables  3  and  4.  (4)  147. 2  pounds.  (66)65.3 
pounds.  (86)  29.38  inches.  (9a)  9.73  kilograms  per  square 
centimeter.  (9d)  223  ooo  kilograms.  (10)  0.0416  horse-powers. 
(13a)  2=1.203  feet.  (14)  See  Art.  180.  (15a)  5670  pounds. 
(166)  TT  inches.  (176)  6.06  feet.  (18)  y  =  \d  for  the  second 
case.  (206)  3.07  for  cement.  (20c)  2945  kilograms.  (2 la) 
8.98  feet  per  second.  (216)  252  feet  per  second.  (24a)  94.6 
and  8.0  feet  per  second.  (25)  #=53.3  feet.  (286)  10.4  horse- 
powers. (32)  22.07  f^et.  (33c)  £  =  0.73  when  q  is  3  cubic 
meters  per  minute.  (36)  0.017  inches.  (38)  14.4  feet  for 
a2.  (426)  Replace  v  by  7^/3.28  and  r  by  7/3.28,  then  the 
new  constant  is  5.97X3.280-431.  (456)^=1.06.  (46)  ^  =  0.99. 
(48a)  {7  =  0.605.  (49a)  17.2  feet.  (50)  103  miner's  inches. 


ART.  195  ANSWERS   TO   PROBLEMS  541 

(52)  2.04  cubic  feet.  (56)  0.122  feet.  (58)  9.4  and  12.3 
square  feet.  (59a)  c1  =  o.gS.  (60)  0.541  feet  per  second. 
(62)-  0.837  feet  per  second1  and  0.0109  feet-  (636)  c  =  0.602. 
(65)  4.039  cubic  feet  per  second.  (66)  7.10  and  6.97  cubic  feet 
per  second.  (67)  21.1  cubic  feet  per  second.  (71)  0.74  percent. 
(72a)  1.30  centimeters.  (746)  0.13  and  7.60  feet.  (756)  0.28 
feet.  (77)^  =  0.985.  (78)  c  =  o.8o2.  (79)  6.67  feet.  (806) 
q  =  0.963.  (82)  0.007  feet-  (85)  °-29  feet-  (86)  Actual  loss 
=  7. 64  feet.  (90a)  3.07  gallons  per  minute.  (91)  3.o6and4.94 
inches.  (93a)  32  pipes.  (936)  11.3  and  8.7  inches,  so  that  12- 
and  9-inch  pipes  should  be  used.  (946)  About  6  cubic  feet  per 
second.  (99)  2.75  feet  per  second;  68  feet.  (1016)  /  =  o.o36. 
(1056)  2.55  feet  per  second.  (107a)  227  cubic  feet  per  second. 
(1076)  4.4  feet.  (108)  1.237  ^eet  an<i  7-32  ^ee^  Per  second. 
(1096)  0.64  feet  deep.  (1106)  5.20  and  3.69  feet  per  second. 
(Ill)  57  400  ooo  gallons.  (113)  6^  =  3.09  feet.  (1196)  0.48  me- 
ters. (121a)  546  cubic  feet  per  second.  (124)  1.59  feet  per 
second.  (126)  761  cubic  feet  per  second.  (127a)  364  pounds. 
(128a)  7.6  feet.  (131a)  ^=12.5  feet.  (133d)  #  =  0.41  meters. 
(137)  13.5  horss-powers.  (138)  1.32  horse-powers.  (139)  257 
feet.  (1406)  35.4  percent.  (142c)  65.800  kilowatts.  (143)  3.96 
gallons.  (146)  93  pounds.  (150)  34.5  feet  per  second.  (151a) 
507.  (1526)  £  =  0.85.  (153a)  ^  =  0.83.  (155)  from  48  to  50 
horse-powers.  (156)  13.6.  (157c)  1.8  horse-powers.  (159s) 
338  revolutions  per  minute.  (162a)  30.1  kilowatts.  (163)  16 
feet.  (1666)  4.117  and  4.120.  (169a)  167.  (170a)  uo°4o'. 
(173<?)  27.0  cubic  meters.  (174)  743  horse-powers.  (1766) 
1530  horse-powers.  (177)  e  is  less  than  o.io.  (182<i)  r=n.6 
meters.  (183a)  76.6  percent.  (1896)  ^  =  0.78.  (191)  17.8 
horse-powers.  (192J)  9 i  meters.  (194/)  See  Hazen's  Filtra- 
tion of  Public  Water  Supplies  (New  York,  1900),  p.  22. 


Evolvi  varia  problemata.  In  scientiis  enim  ediscendis  pro- 
sunt  exempla  magis  quam  pra3cepta.  Qua  de  causa  in  his  fusius 
expatiatus  sum.  NEWTON. 


542  APPENDIX  ART.  19$ 

ART.  196.      EXPLANATION  OF  TABLES 

The  following  hydraulic  tables  have  been  mostly  ex- 
plained in  the  text,  and  at  the  foot  of  each  table  a  reference 
is  given  to  the  article  where  the  explanation  may  be  found. 

Table  50  gives  squares  of  numbers  from  i.oo  to  9.99,. 
the  arrangement  being  the  same  as  that  of  the  logarithmic 
table.  By  properly  moving  the  decimal  point,  four-place 
squares  of  other  numbers  are  also  readily  taken  out.  For 
example,  the  square  of  0.874  is  0.7639,  and  that  of  87.4  is 
7639,  correct  to  four  significant  figures.  This  table  may  also 
be  used  for  finding  square  roots  of  numbers. 

Table  51  gives  areas  of  circles  for  diameters  ranging 
from  i.oo  to  9.99,  arranged  in  the  same  manner,  and  by 
properly  moving  the  decimal  point,  four-place  areas  for  all 
circles  can  be  found.  For  instance,  if  the  diameter  is  4.175 
inches,  the  area  is  13.69  square  inches;  if  the  diameter  is 
0.535  feet,  the  area  is  0.2248  square  feet.  This  table  may 
also  be  used  for  finding  diameters  of  circles  corresponding 
to  given  areas. 

Table  52  gives  trigonometric  functions  of  angles  and 
Table  53  the  logarithms  of  these  functions.  The  term  "  arc" 
means  the  length  of  a  circular  arc  of  radius  unity,  while 
"  coarc  "  is  the  complement  of  the  arc,  or  a  quadrant  minus 
the  arc.  If  6  be  the  number  of  degrees  in  any  angle,  the 
value  of  arc0  is  nd/i8o. 

Table  54  gives  four-place  common  logarithms  of  num- 
bers, and  these  are  of  great  value  in  hydraulic  computations 
(Art.  8).  Table  55,  taken  from  the  author's  "  Elements  of 
Precise  Surveying  and  Geodesy,"  gives  mathematical  con- 
stants and  their  logarithms  to  nine  decimals ;  this  is  a  greater 
number  than  will  ever  be  needed  in  hydraulic  work,  but 
they  are  sometimes  required  for  the  discussion  of  geodetic 
and  physical  measurements. 


HYDRAULIC  TABLES 


543 


TABLE  1.     FUNDAMENTAL  HYDRAULIC  CONSTANTS 
English  Measures 


Name 

Symbol 

Number 

Logarithm 

Pounds  of  water  in  one  cubic  foot 

•w 

62.5 

i  .  7959 

Pounds  of  water  in  one  U.  S.  gallon 

70/7.481 

8-355 

0.9220 

Pounds   per   square  inch    due  to  one 
atmosphere 

14-7 

1.1673 

Pounds  per  square  inch  due  to  one  foot 
of  head 

w/144 

0-434 

1.6375 

Feet  of  head  for  pressure  of  one  pound 
per  square  inch 

144/w 

2.304 

0.3625 

Cubic  feet  in  one  U.  S.  gallon 

231/1728 

0.1337 

T.  1261 

U.  S.  gallons  in  one  cubic  foot 

1728/231 

7.481 

0.8739 

Acceleration    of    gravity    in    feet  per 
second  per  second 

g 

32.16 

i  •  5073 

\/2g 

8.O2O 

o  .  9042 

$V2g 

5.347 

0.7281 

I/2g 

0.01555 

2  .  1916 

\K\/2g 

6.299 

0.7993 

Explanation  in  Arts.  4-11. 


TABLE  2.     FUNDAMENTAL  HYDRAULIC  CONSTANTS 
Metric  Measures 


Name 

Symbol 

Number 

Logarithm 

Kilograms  of  water  in  one  cubic  meter 

TV 

IOOO 

3.0000 

Kilograms  of  water  in  one  liter 

W/IOOO 

I 

o.ocoo 

Kilograms  per  square  centimeter  due 
to  one  atmosphere 

1.033 

0.0142 

Kilograms  per  square  centimeter  due 
to  one  meter  head 

W/IOOOO 

O.I 

T.oooo 

Meters  of  head  for  pressure  of  one 
kilogram  per  square  centimeter 

IOOOO/W 

10 

I.  0000 

Cubic  meters  in  one  liter 

I/IOOO 

0.001 

^.0000 

Liters  in  one  cubic  meter 

IOOO/I 

IOOO 

3.0000 

Acceleration  of  gravity  in  meters  per 
second  per  second 

g 

9.800 

0.9912 

\/2g 

4.427 

0.6461 

fv^ 

2.951 

0.4700 

I/2g 

0.05102 

2.7077 

i*\/2g 

3-477 

0.5412 

Explanation  in  Arts.  9  and  20. 


544  HYDRAULIC  TABLES 

TABLE  3.     METRIC  EQUIVALENTS  OF  ENGLISH  UNITS 


EngHsh  Unit 

Metric  Equivalent 

Logarithm 

i  Inch 

2.5400  centimeters 

0.40483 

i  Foot 

0.3048  meters 

I  .  48402 

Square  Inch 

6.4520  square  centimeters 

o  .  80969 

Square  Foot 

0.09290  square  meters 

2  .  96803 

Cubic  Foot 

0.02832  cubic  meters 

2.45209 

U.  S.  Gallon 

3  .  7854  liters 

0.57812 

Imperial  Gallon 

4  .  5438  liters 

0.65742 

Pound 

0.4536  kilograms 

1.65667 

Pound      per     Square 
Inch 

0.07030     kilograms    per     square 
centimeter 

2.84697 

Pound  per  Cubic  Foot 

16.017  kilograms  per  cubic  meter 

1.20457 

Foot-pound 

o.  1383  kilogram-meters 

I.  14069 

Horse-power 

i  .0139  horse-powers 

0.00599 

Fahrenheit 

Centigrade  temperature 

Temperature  F° 

C0=!(F°-32°) 

TABLE  4.     ENGLISH  EQUIVALENTS  OF  METRIC  UNITS 


Metric  Unit 

English  Equivalent 

Logarithm 

Centimeter 

o  .  3937  inches 

^•59517 

Meter 

3  .  2808  feet 

0.51598 

Square  Centimeter 

o  .  1550  square  inches 

1.19031 

Square  Meter 

10.  764  square  feet 

1.03197 

Cubic  Meter 

35.  3  14  cubic  feet 

I-5479I 

Liter 

0.2642  U.  S.  gallons 

1.42188 

Liter 

0.2201  imperial  gallons 

1.34258 

Kilogram 

2.2046  pounds 

0-34333 

Kilogram  per  Square 
Centimeter 

14.224  pounds  per  square  inch 

I.I5303 

i  Kilogram   per   Cubic 
Meter 

0.06244  pounds  per  cubic  foot 

2-79543 

i  Kilogram-meter 

7.2329  foot-pounds 

0.85931 

i  Horse-power 

o  .  9863  horse-powers 

1.99041 

Centigrade 

Fahrenheit  temperature 

Temperature  C° 

F°  =  32°+i.8C0 

HYDRAULIC  TABLES 


545 


TABLE  5.     INCHES  REDUCED  TO  FEET 
English  Measures 


Inches 

Feet 

Inches 

Feet 

Square 
Inches 

Square 
Feet 

Cubic 
Inches 

Cubic 
Feet 

i 

0.0104 

3 

0.2500 

IO 

0.6944 

1000 

0.5787 

* 

.0208 

4 

•3333 

20 

1.3889 

2000 

I-I574 

i 

•  0313 

5 

.4167 

30 

2.0833 

3000 

I.736I 

* 

.0417 

6 

.5000 

40 

2.6777 

4000 

2.3H8 

i 

.0521 

7 

.5833 

50 

3.4722 

5000 

2.8935 

i 

.0625 

8 

.6667 

60 

4.1667 

6OOO 

3.4722 

i 

.0729 

9 

.7500 

70 

4-5500 

7000 

4.0509 

i 

•0833 

10 

.8333 

80 

5-3555 

8000 

4.6296 

2 

.1667 

ii 

.9167 

90 

6.2500 

9OOO 

5  •  2083 

Explanation  in  Art.  8. 


TABLE  6.     GALLONS  AND  CUBIC  FEET 


English  Measures 


Cubic 
Feet 

U.S. 
Gallons 

U.  S. 
Gallons 

Cubic 
Feet 

Cubic 
Feet 

Imperial 
Gallons 

Imperial 
Gallons 

Cubic 
Fe«t 

I 

7.481 

I 

0.1337 

I 

6.232 

I 

o.  16046 

2 

14.961 

2 

0.2674 

2 

12.464 

2 

0.3209 

3 

22.442 

3 

0.4010 

3 

18.696 

3 

0.4814 

4 

28.922 

4 

0-5347 

4 

24.928 

4 

0.6418 

5 

37.403 

5 

0.6684 

5 

32.160 

5 

0.8023 

6 

44.883 

6 

0.8021 

6 

37-393 

6 

0.9628 

7 

52-364 

7 

0.9358 

7 

43-625 

7 

1.1232 

8 

59.844 

8 

1.0695 

8 

49.857 

8 

1.2837 

9 

67.325 

9 

1.2031 

9 

56.089 

9 

1.4442 

10 

74.805 

10 

1.3368 

10 

62.321 

IO 

i  .  6046 

Explanation  in  Art.  8. 


546 


HYDRAULIC  TABLES 


TABLE  7.     WEIGHT  OF  DISTILLED  WATER 

English  Measures 


Temperature 
Fahrenheit 

Pounds 
per 
Cubic  Foot 

Temperature 
Fahrenheit 

Pounds 
per 
Cubic  Foot 

Temperature 
Fahrenheit 

Pounds 
per 
Cubic  Foot 

32° 

62.42 

95 

62.06 

160 

61.01 

35 

62.42 

100 

62.OO 

165 

60.90 

39-3 

62  .424 

105 

61.93 

170 

60.80 

45 

62.42 

no 

61.86 

i?5 

60.69 

50 

62.41 

115 

61.79 

1  80 

60.59 

55 

62.39 

120 

61  .72 

185 

60.48 

60 

62.37 

125 

61.64 

190 

60.36 

65 

62.34 

130 

6i.55 

195 

60.25 

70 

62.30 

135 

61.47 

200 

60.  14 

75 

62  .  26  ' 

I4O 

61-39 

205 

60.02 

80 

62.22 

145 

61.30 

210 

59.89 

85 

62.17 

150 

61.20 

212 

59.84 

90 

62.  12 

155 

6i.n 

Explanation  in  Art.  4. 


TABLE  8.     WEIGHT  OF  DISTILLED  WATER 
Metric  Measures 


Temperature 

Kilograms 

Temperature 

Kilograms 

Temperature 

Kilograms 

Centigrade 

per 
Cubic  Meter 

Centigrade 

per 
Cubic  Meter 

Centigrade 

per 
Cubic  Meter 

-3° 

999-59 

1  6° 

999.00 

55° 

985.85 

0 

999.87 

18 

998.65 

60 

983-38 

+  3 

999-99 

20 

998.26 

65 

980.74 

4 

1000. 

22 

997.83 

70 

977-94 

5 

999  .  99     - 

25 

997.12 

75 

974.98 

6 

999  •  97 

30 

995  -  76 

80 

971-94 

8 

999.89 

35 

994-13 

85 

968.79 

10 

999-75 

40 

992  -  35 

90 

965-56 

12 

999-55 

45 

990.37 

95 

962.19 

14 

999.30 

50 

998.20 

IOO 

958.65 

Explanation  in  Art.  9. 


HYDRAULIC  TABLES 


547 


TABLE  9.     ATMOSPHERIC  PRESSURE 

English  Measures 


Mercury 
Barometer 
Inches 

Pressure 
Pounds  per 
Square  Inch 

Pressure 
Atmospheres 

Water 
Barometer 
Feet 

Elevations 
Feet 

Boiling-point 
of  Water 

Fahrenheit 

31 

15-2 

1.03 

35-1 

—  890 

213°.  9 

30 

14-7 

I. 

34-0 

0 

212    .2 

29 

14.2 

0.97 

32.9 

+  920 

210    .4 

28 

13-7 

0-93 

31.7 

1880 

208  .7 

27 

13-2 

0.90 

30.6 

2870 

206    .9 

26 

12.7 

0.86 

29.5 

3900 

205  .0 

25 

12.2 

0.83 

28.3 

4970 

203    .1 

24 

II-  7 

0.80 

27.2 

6080 

2OI    .  I 

23 

ii.  3 

0.76 

26.1 

7240 

199    -0 

22 

10.8 

0.72 

24.9 

8455 

196    .9 

21 

10.3 

0.69 

23.8 

9720 

194  -7 

20 

9.8 

0.67 

22.7 

II050 

192  .4 

Explanation  in  Art.  5. 


TABLE  10.     ATMOSPHERIC  PRESSURE 

Metric  Measures 


Mercury 
Barometer 

Pressure 
Kilograms 

Pressure 

Water 
Barometer 

Elevations 

Boiling-point 
of  Water 

Millimeters 

per  Square 
Centimeter 

Atmospheres 

Meters 

Meters 

Centigrade 

790 

1.074 

1  .04 

10.74 

-325 

101°.  I 

760 

1-033 

I. 

10-33 

0 

100    . 

730 

0.992 

0.96 

9.92 

+  340 

98  .9 

700 

•952 

•92 

9-52 

690 

97  -8 

670 

.911 

.88 

9.II 

1045 

96  .6 

640 

.870 

.84 

8.70 

1420 

95  -4 

610 

.829 

.80 

8.29 

1820 

94  -i 

580 

.788 

.76 

7.88 

2240 

92  .8 

550 

.748 

.72 

7.48 

2680 

9i  -5 

520 

.707 

.68 

7-07 

3140 

90  .1 

Explanation  in  Art.  9. 


548 


HYDKAULIC  TABLES 


TABLE  11.     ACCELERATION  OP  GRAVITY 
English  Measures 


No. 

Multiples 
of  g 

Multiples 
of  2g 

Multiples 
of  i/2g 

Multiples 

ofv^T 

No. 

I 

32.16 

64.32 

0.01555 

8.02 

I 

2 

64.32 

128.6 

0.03109 

16.04 

2 

3 

96.48 

193.0 

0.04664 

24.06 

3 

4 

128.6 

257-3 

0.06219 

32.08 

4 

5 

160.8 

321.6 

0.07774 

40.  10 

5 

6 

193.0 

385.9 

0.09328 

48.12 

6 

7 

225.1 

450.2 

0.1088 

56.14 

7 

8 

257-3 

5H.5 

o  .  i  244 

64.16 

8 

9 

289.4 

578.9 

0.1399 

72.18 

9 

10 

321.6 

643.2 

0.1555 

80.20 

10 

Explanation  in  Art.  7- 


TABLE  12.     ACCELERATION  OF  GRAVITY 
Metric  Measures 


No. 

Multiples 
of  g 

Multiples 
of  sg 

Multiples 

Of   I/2g 

Multiples 
of  \/2g 

No. 

I 

9.800 

19.60 

0.05102 

4.427 

I 

2 

19.60 

39-20 

O.  IO2O 

8.854 

2 

3 

29.40 

58.80 

O.I53I 

13.282 

3 

4 

39-20 

78.40 

0.2041 

17.71 

4 

5 

49.00 

98.00 

0.2551 

22  .  14 

5 

6 

58.80 

117.60 

0.3061 

26.56 

6 

7 

68.60 

137.2 

0.3571 

30.99 

7 

8 

78.40 

156.8 

0.4082 

35.42 

8 

9 

88.20 

176.4 

0.4592 

39.84 

9 

10 

98.00 

196.0 

0.5102 

44-27 

10 

Explanation  in  Art.  9- 


HYDRAULIC  TABLES 


549 


TABLE  13.  HEADS  AND  PRESSURES 

English  Measures 


Head 

in  Feet 

Pressure  in  Pounds 
per  Square  Inch 

Pressure 
in  Pounds 
per 
Square 
Inch 

Head  in  Feet 

«;  =  62.5 

a>  =  62.3 

«/  =  62.5 

70  =  62.3 

I 

0-434 

0-433 

I 

2.304 

2.311 

2 

0.868 

0.865 

2 

4.608 

4.623 

3 

1.302 

1.298 

3 

6.912 

6-934 

4 

1.736 

I.73I 

4 

9.216 

9.246 

5 

2.170 

2.163 

5 

11.520 

11-557 

6 

2.604 

2.596 

6 

13.824 

13.868 

7 

3.038 

3-028 

7 

16.128 

16.180 

8 

3-472 

3.46I 

8 

18.432 

18.491 

9 

3.906 

3.894 

9 

20.736 

20.803 

10 

4.340 

4.326 

10 

23.040 

23.114 

Explanation  in  Art.  11. 


TABLE  14.     HEADS  AND  PRESSURES 

Metric  Measures 


Head 
in  Meters 

Pressure  in  Kilograms 
per  Square  Centimeter 

Pressure 
in  Kilo- 
grams per 
Square 
Centim'ter 

Head  in  Meters 

w  =•  1000 

u;  =  997 

w  =  1000 

u;  =  997 

I 

O.I 

0.0997 

I 

IO 

10.03 

2 

O.2 

0.1994 

2 

20 

20.06 

3 

0-3 

o.  2991 

3 

30 

30.09 

4 

0.4 

0.3988 

4 

40 

40.12 

5 

0.5 

0.4985 

5 

50 

50.15 

6 

0.6 

0.5982 

6 

60 

60.  18 

7 

0.7 

0.6979 

7 

70 

70.21 

8 

0.8 

0.7976 

8 

80 

80.24 

9 

0.9 

0.8973 

9 

90 

90.27 

10 

I.O 

0.9970 

10 

100 

100.30 

Explanation  in  Art.  20. 


550 


HYDRAULIC  TABLES 


TABLE  15.     VELOCITIES  AND  VELOCITY-HEADS 

English  Measures 


iV-Va2P-*'Oao>/5 

/t=V2/2g  =  o.oisssF2 

Head 
in  Feet 

Velocity 
in  Feet 
per  Second 

Head 
in  Feet 

Velocity 
in  Feet 
per  Second 

Velocity 
in  Feet 
per 
Second 

Head 
in  Feet 

Velocity 
in  Feet 
per 
Second 

Head 
in  Feet 

O.I 

2-537 

I 

8.02 

I 

0.016 

10 

1.56 

0.2 

3-587 

2 

"•33 

2 

0.062 

20 

6.22 

0.3 

4-393 

3 

13.89 

3 

0.  140 

30 

13.99 

0.4 

5.072 

4 

16.04 

4 

0.249 

40 

24.88 

0.5 

5-671 

5 

17-93 

5 

0.389 

50 

36.86 

0.6 

6.212 

6 

19.64 

6 

0.560 

60 

55-97 

0.7 

6.710 

7 

21.22 

7 

0.762 

70 

76.19 

0.8 

7.I7I 

8 

22.68 

8 

0.995 

80 

99-51 

0.9 

7.608 

9 

24.06 

9 

1.260 

90 

125.95 

I.O 

8.020 

10 

25.36 

10 

1-555 

100 

155.50 

Explanation  in  Art.   22. 

TABLE  16.     VELOCITIES  AND  VELOCITY-HEADS 
Metric  Measures 


V=  \/2gF=  4-427  \/h 

k=  V2/2g=0.0$l02  V2 

Head  in 
Meters 

Velocity 
in  Meters 
per  Second 

Head  in 
Meters 

Velocity 
in  Meters 
per  Second 

Velocity 
in 
Meters 
per 
Second 

Head  in 
Meters 

Velocity 
in 
Meters 
per 
Second 

Head  in 
Meters 

O.I 

1.432 

I 

4.427 

O.I 

0.0005 

I 

0.0510 

0.2 

1.980 

2 

6.262 

O.2 

O.OO2O 

2 

o.  2041 

0.3 

2.425 

3 

7.668 

0-3 

0.0046 

3 

0.4592 

0.4 

2-799 

4 

8.854 

0-4 

O.OO82 

4 

0.8163 

0.5 

3.I3I 

5 

9.900 

0-5 

0.0123 

5 

1  .  276 

0.6 

3.429 

6 

10.84 

0.6 

0.0184 

6 

1.837 

0.7 

3.704 

7 

11.71 

0.7 

O.O25O 

7 

2.500 

0.8 

3.960 

8 

12.52 

0.8 

0.0327 

8 

3.265 

0.9 

4.200 

9 

13.28 

0-9 

0.0413 

9 

4.133 

i  .0 

4.427 

10 

14.00 

i  .0 

O.O5IO 

10 

5-102 

Explanation  in  Art.  33. 


HYDRAULIC  TABLES 


551 


TABLE  17.     COEFFICIENTS  FOR  CIRCULAR  VERTICAL  ORIFICES 

Arguments  in  English  Measures 


Head 
h 
in  Feet 

Diameter  of  Orifice  in  Feet 

0.02 

0.04 

0.07 

O.I 

0.2 

0.6 

I  .0 

0.4 

0.637 

0.624 

0.618 

0.6 

0.655 

.630 

.618 

•613 

0.601 

0.593 

0.8 

.648 

.626 

.615 

.610 

.601 

•  594 

0.590 

1.0 

.644 

.623 

.612 

.608 

.600 

•  595 

.591 

1-5 

.637 

.618 

.608 

.605 

.600 

.596 

.593 

2. 

.632 

.614 

.607 

.604 

•599 

•  597 

•595 

2.5 

.629 

.612 

.605 

.603 

•599 

.598 

.596 

3- 

.627 

.611 

.604 

.603 

.599 

.598 

.597 

4- 

.623 

.609 

.603 

.602 

•599 

•  597 

•  596 

6. 

.618 

.607 

.602 

.600 

•  598 

•  597 

.596 

8. 

.614 

.605 

.601 

.600 

.598 

.596 

.596 

10. 

.611 

.603 

•599 

.598 

•  597 

.596 

•  595 

20. 

.601 

•  599 

.597 

.596 

.596 

.596 

•  594 

50. 

.596 

•  595 

•  594 

•  594 

•  594 

•  594 

•  593 

IOO. 

•  593 

.592 

•  592 

.592 

•  592 

•  592 

•  592 

Explanation  in  Art.  47. 


TABLE  18.     COEFFICIENTS  FOR  CIRCULAR  ORIFICES 

Arguments  in  Metric  Measures 


Head 
h 
in  Meters 

Diameter  of  Orifice  in  Centimeters 

i 

2 

3 

6 

18 

30 

O.  I 

0.642 

0.626 

0.619 

0.2 

.639 

.619 

.613 

0.601 

0.593 

0-3 

.634 

.613 

.608 

.600 

.595 

o.59i 

0.5 

.626 

.609 

.605 

.600 

•  596 

•  593 

0.7 

.620 

.607 

•  603 

•599 

•  598 

•596 

I  . 

.619 

.605 

.602 

•599 

.598 

•  597 

1.5 

.614 

.6O4 

.601 

.598 

•  597 

•  596 

2. 

.611 

.603 

.600 

•597 

•  596 

•  596 

3- 

.607 

.600 

.598 

•597 

••596 

•  595 

6. 

.600 

.597 

.596 

.596 

•  596 

•  594 

15- 

.596 

•  595 

•594 

•594 

•  594 

•593 

30. 

•593 

•  592 

•592 

•  592 

•  592 

•  592 

Explanation  in  Art.  59. 


552 


HYDRAULIC  TABLES 


TABLE  19.     COEFFICIENTS  FOR  SQUARE  VERTICAL  ORIFICES 
Arguments  in  English  Measures 


Head 
h 

in  Feet 

Side  of  the  Square  in  Feet 

0.02 

0.04 

0.07 

0.  I 

0.  2 

0.6 

I  .  O 

0.4 

0.643 

0.628 

0.621 

0.6 

0.660 

.636 

.623 

.617 

0.605 

0.598 

0.8 

.652 

.631 

.620 

.615 

.605 

.600 

°-597 

i  .0 

.648 

.628 

.618 

.613 

.605 

.601 

.599 

1.5 

.641 

.622 

\ 

.614 

.610 

.605 

.602 

.601 

2. 

•637 

.619 

.612 

.608 

.605 

.604 

.602 

2.5 

.634 

.617 

.610 

.607 

•605 

.604 

.602 

3- 

.632 

.616 

.609 

.607 

•60S 

.604 

.603 

4- 

.628 

.614 

.608 

.606 

.605 

.603 

.602 

6. 

.623 

.612 

.607 

.605 

.604 

•603 

.602 

8. 

.619 

.610 

.606 

.605 

.6O4 

.603 

.602 

10. 

.616 

.608 

.605 

.604 

.603 

.602 

.601 

20. 

.606 

.604 

.602 

.602 

.602 

.601 

.600 

50. 

.602 

.601 

.601 

.600 

.600 

•599 

•599 

100. 

•  599 

.598 

.598 

.598 

.598 

•598 

.598 

Explanation  in  Art.  48. 

TABLE  20.     COEFFICIENTS  FOR  SQUARE  VERTICAL  ORIFICES 
Arguments  in  Metric  Measures 


Head 
k 

in  Meters 

Side  of  the  Square  in  Centimeters 

i 

2 

3 

6 

I  2 

30 

O.  I 
»          0.2 
0-3 
0-5 

0.7 
1  .0 
i.5 

2. 

3- 
6. 

IS- 
30- 

0.652 
.648 
.636 
.628 
.625 
.620 
.618 
.614 
.611 
.605 
.601 
.598 

0.632 
.624 
.619 
.6l8 
.612 
.6lO 
.609 
.608 
.606 
.603 
.6OI 
.598 

O.622 
.617 
.613 
.6lO 
.607 
.607 
.606 
.605 
.604 
.602 
.6OO 
.598 

0.605 

0.598 
.601 
.6O2 

0-599 
.601 
.602 
.603 

0.605 
.605 
.605 
.605 
.604 
.604 
.603 
.602 
.600 
.598 

.604 
.604 
.603 
.603 
.602 
.601 

•599 
.598 

.602 
.602 
.601 
.600 
•  599 
.598 

Explanation  in  Art.  59. 


HYDRAULIC  TABLES 


TABLE  21.     COEFFICIENTS  FOR  RECTANGULAR  ORIFICES 

i  FOOT  WIDE 
Arguments  in  English  Measures 


Head 
h 
in  Feet 

Depth  of  Orifice  in  Feet 

0.125 

0.25 

0.50 

o.7S 

I  .0 

i.S 

2.0 

0.4 

0.634 

0.633 

0.622 

0.6 

•633 

.633 

.619 

0.614 

0.8 

.633 

.633 

.618 

.612 

0.6o8 

i. 

.632 

.632 

.618 

.612 

.606 

0.626 

i-5 

.630 

•631 

.618 

.611 

.605 

.626 

0.628 

2. 

.629 

.630 

.617 

.611 

.605 

.624 

.630 

2-5 

.628 

.628 

.616 

.611 

.605 

.616 

.627 

3- 

.627 

.627 

.615 

.6lO 

.605 

.614 

.619 

4- 

.624 

.624 

.614 

.609 

.605 

.612 

.6l6 

6. 

.615 

.615 

.609 

.604 

.602 

.606 

.610 

8. 

.609 

.607 

.603 

.602 

.601 

.602 

.604 

10. 

.606 

.603 

.601 

.601 

.601 

.601 

.602 

20. 

.601 

.601 

.601 

.602 

Explanation  in  Art.  49. 


TABLE  22.     COEFFICIENTS  FOR  SUBMERGED  ORIFICES 
Arguments  in  English  Measures 


Size  of  Orifice  in  Feet 

Effective 

Head  in 

Feet 

Circle 

Square 

Circle 

Square 

Rectangle 

0.05 

0.05 

0.  I 

0.  I 

0.05X0.3 

0.5 

0.615 

0.619 

0.603 

0.608 

0.623 

1  .0 

.610 

.614 

.602 

.606 

.622 

1-5 

.607 

.612 

.600 

.605 

.621 

2.0 

.605 

,6lO 

•599 

.604 

.620 

2-5 

•  603 

.608 

•598 

.604 

.619 

3-0 

.602 

.607 

.598 

.604 

.618 

4.0 

.601 

.606 

.598 

.604 

Explanation  in  Art.  52. 


554 


HYDRAULIC  TABLES 


TABLE  23.     COEFFICIENTS  FOR  CONTRACTED  WEIRS 

Arguments  in  English  Measures 


Effective 
Head 

Length  of  Weir  in  Feet 

in  Feet 

0.66 

i 

2 

3 

5 

10 

19 

0.  I 

0.632 

0.639 

0.646 

0.652 

0.653 

0.655 

0.656 

0.15 

.619 

.625 

.634 

.638 

.640 

.641 

.642 

O.  2 

.611 

.618 

.626 

.630 

.631 

.633 

.634 

0.25 

.605 

.612 

.621 

.624 

.626 

.628 

.629 

0-3 

.601 

.608 

.6l6 

.619 

.621 

.624 

.625 

0.4 

•  595 

.601 

.609 

.613 

.615 

.618 

.620 

0-5 

•  590 

.596 

.605 

.608 

.611 

.615 

.617 

0.6 

.587 

•593 

.601 

.605 

.608 

.613 

.615 

0.7 

•  590 

.598 

.603 

.606 

.612 

.614 

0.8 

•  595 

.6OO 

.604 

.611 

.613 

0-9 

*" 

•  592 

.598 

.603 

.609 

.612 

i  .0 

•  590 

•  595 

.601 

.608 

.611 

I  .2 

.585 

•  591 

•  597 

.605 

.610 

i-4 

.580 

•  587 

•  594 

.602 

.609 

1.6 

.582 

•  591 

.600 

.607 

Explanation  in  Art.  63. 

TABLE  24.     COEFFICIENTS  FOR  CONTRACTED  WEIRS 

Arguments  in  Metric  Measures 


Effective 
Head  in 

Centi- 
meters 

Length  of  Weir  in  Meters 

O.2 

0.3 

0.6 

0.9 

i.  5 

3.0 

5-8 

3- 

0.633 

0.640 

0.647 

0.653 

0.654 

0.656 

0.657 

5- 

.618 

.624 

.634 

.638 

.640 

.641 

.642 

7- 

.606 

.613 

.622 

.625 

.627 

.629 

.630 

9- 

.601 

.608 

.616 

.619 

.621 

.624 

.625 

12. 

.596 

.602 

.609 

•613 

.615 

.618 

.620 

15- 

•591 

•  597 

.605 

.608 

.611 

.615 

.617 

18. 

.588 

•  593 

.601 

.605 

.608 

•613 

.615 

22. 

.589 

•597 

.603 

.606 

.612 

.614 

26. 

•594 

•  599 

.604 

.610 

.613 

13°- 

•  590 

•  595 

.601 

.608 

.611 

35- 

.586 

•  592 

•  597 

.605 

.6lO 

45- 

.585 

•593 

.601 

.608 

Explanation  in  Art.  72. 


HYDRAULIC  TABLES 


555 


TABLE  25.     COEFFICIENTS  FOR  SUPPRESSED  WEIRS 
Arguments  in  English  Measures 


Effective 
Head 
in  Feet 

Length  of  Weir  in  Feet 

19 

10 

7 

5 

4 

3 

2 

O.I 

0.657 

0.658 

0.658 

0.659 

0.15 

.643 

.644 

.645 

.645 

0.647 

0.649 

0.652 

0.2 

.635 

.637 

.637 

.638 

.641 

.642 

.645 

0.25 

.630 

•632 

.633 

.634 

.636 

.638 

.641 

0-3 

.626 

.628 

.629 

.631 

•633 

•  636 

.639 

0.4 

.621 

.623 

.625 

.628 

.630 

.633- 

.636 

0-5 

.619 

.621 

.624 

.627 

.630 

.633 

.637 

0.6 

.618' 

.620 

.623 

.627 

.630 

.634 

.638 

0.7 

.618 

.620 

.624 

.628 

.631 

.635 

.640 

0.8 

.618 

.621 

.625 

.629 

•633 

.637 

.643 

0.9 

.619 

.622 

.627 

.631 

.635 

.639 

.645 

I.O 

.619 

.624 

.628 

.633 

.637 

.641 

.648 

I  .2 

.620 

.626 

.632 

.636 

.641 

.646 

i-4 

.622 

.629 

.634 

.640 

.644 

1.6 

.623 

.631 

.637 

.642 

•647 

Explanation  in  Art.  64. 

TABLE  26.     COEFFICIENTS  FOR  SUPPRESSED  WEIRS 
Arguments  in  Metric  Measures 


Effective 
Head  in 
Centi- 
meters 

Length  of  Weir  in  Meters 

5.8 

3.o 

2.0 

i.S 

I  .  2 

o.9 

0.6 

3- 

0.658 

0.659 

0.659 

0.660 

5- 

.642 

.643 

.644 

.645 

0.647 

0.649 

0.652 

7- 

•632 

.633 

•634 

•635 

.637 

.640 

.643 

9- 

.626 

.628 

.629 

.631 

.633 

•  636 

•639 

12. 

.621 

.623 

.625 

.628 

.630 

.633 

.636 

15- 

.619 

.621 

.624 

.627 

.630 

•633 

.637 

18. 

.618' 

.620 

.623 

.627 

.630 

•634 

.638 

22. 

.618 

.620 

.624 

.628 

.632 

.636 

.640 

26. 

.619 

.622 

.627 

.631 

.635 

.639 

.645 

30. 

.619 

.624 

.628 

.633 

.637 

.641 

35- 

.620 

.626 

.631 

.635 

.640 

.645 

45- 

.622 

.630 

.635 

.641 

.645 

Explanation  in  Art.  72. 


556 


HYDRAULIC  TABLES 


TABLE  27.     FACTORS  FOR  SUBMERGED  WEIRS 
For  all  Measures 


H' 
~H 

n 

H' 
H 

n 

H' 
H 

n 

H' 
H 

n 

0.00 

I  .OOO 

0.18 

0.989 

0.38 

0.935 

0.58 

0.856 

.01 

1.004 

.20 

0.985 

.40 

0.929 

.60 

0.846 

.02 

I  .006 

.22 

0.980 

.42 

0.922 

.62 

0.836 

.04 

1.007 

•24 

0-975 

•44 

0-915 

.64 

0.824 

.06 

I  .007 

.26 

0.970 

-46 

0.908 

.66 

0.813 

.08 

I  .006 

.28 

0.964 

-48 

0.900 

.70 

0.787 

.  IO 

1.005 

•30 

0-959 

•50 

0.892 

-75 

0.750 

.  12 

I  .OO2 

•32 

0.953 

-52 

0.884 

.80 

0.703 

•14 

0.998 

•  34 

0-947 

•  54 

0.875 

.90 

0-574 

.16 

0.994 

.36 

0.941 

-56 

0.866 

i  .00 

0.000 

Explanation  in  Art.  66. 


TABLE  28.     CORRECTIONS  FOR  WIDE  CRESTS 

English  Measures 


Head  on 
Wide 
Crest 
Feet 

Width  of  Crest  in  Inches 

2 

4 

6 

8 

10 

12 

24 

0.05 

0.010 

O.OO9 

O.OO9 

O.OO9 

0.009 

0.009 

O.OO9 

.  IO 

.016 

.018 

.017 

.017 

.017 

.017 

.017 

.20 

.012 

.029 

.031 

.032 

.033 

•033 

•034 

-30 

.030 

.041 

•045 

.047 

.048 

.050 

.40 

.022 

-045 

-055 

.060 

.062 

.066 

-50 

.006 

.041 

.060 

.069 

.074 

.082 

.60 

.031 

•°59 

.075 

.083 

.097 

.70 

.017 

.052 

.075 

.089 

.  112 

.80 

.000 

.040 

.071 

'.091 

•125 

.90 

.027 

.062 

.089 

•137 

1.  00 

.on 

.050 

.082 

.149 

1.20 

.021 

.061 

.168 

I  .40 

.032 

.180 

Explanation  in  Art.  67. 


HYDRAULIC  TABLES 


'   557 


TABLE  29.     COEFFICIENTS  FOR  DAMS 
English  Measures 


Up- 
Stream 
Slope 

Width 
of  Crest 
Feet 

Down- 
Stream 
Slope 

Head  H  on  Crest  in  Feet 

o.S 

I  .0 

i.S 

2  .0 

3.o 

4.o 

5.0 

i  on  2 

0.33 

Vertical 

3-35 

3-68 

3-82 

3-77 

3-68 

3-70 

3-71 

i  on  2 

0.66 

Vertical 

3.22 

3-44 

3-59 

3-66 

3-68 

3.70 

3-71 

i  on  5 

0.66 

Vertical 

3-31 

3-33 

3-34 

3-35 

3.38 

3-39 

3-39 

i  on  4 

0.66 

Vertical 

3-44 

3.46 

3.48 

3.48 

3.48 

3.48 

i  on  3 

0.66 

Vertical 

3-64 

3-82 

3.83 

3-69 

3-55 

3-55 

3-55 

i  on  2 

o.oo 

i  on  i 

4.21 

4.24 

4.09 

3-97 

3.83 

3-74 

3-68 

i  on  2 

0.66 

i  on  2 

3-*4 

3-42 

3-45 

3-61 

3-66 

3-66 

3-64 

i  on  2 

0-33 

i  on  5 

3-30 

3-57 

3-60 

3-51 

3-47 

3-54 

3-57 

Vertical 

2.62 

Vertical 

2.6b 

2.67 

2.75 

2.84 

3.01 

3.21 

3-39 

Vertical 

2.62* 

Vertical 

2.96 

3.01 

3-03 

3-08 

3-25 

3.38 

3-47 

Vertical 

6.56 

Vertical 

2.50 

2.60 

2-54 

2.48 

2.51 

2.61 

2.70 

Vertical 

6.56* 

Vertical 

2.71 

2.83 

2.84 

2.84 

2.86 

2.90 

2.94 

i  on  i 

Round 

Vertical 

2.95 

3-17 

3.3i 

3-45 

3.56 

3-6i 

3.65 

Explanation  in  Art.  68. 

TABLE  30.     COEFFICIENTS  FOR  DAMS 
Metric  Measures 


Up- 
Stream 
Slope 

Width 
of  Crest 
Meters 

Down- 
Stream 
Slope 

Head  H  on  Crest  in  Meters 

0.15 

0.30 

0.60 

O.QI 

I  .  22 

1.52 

I  on  2 

0.10 

Vertical 

1.85 

2.03 

2.08 

2.03 

2.04 

2.05 

i  on  2 

O.2O 

Vertical 

1.78 

1.90 

2.02 

2.03 

2.04 

2.05 

i  on  5 

O.2O 

Vertical 

1-83 

1.84 

1.85 

1.86 

1.87 

1.87 

i  on  4 

0.20 

Vertical 

1.90 

1.92 

1.92 

1.92 

1.92 

i  on  3 

O.  2O 

Vertical 

2.OI 

2.  II 

2.O4 

1.96 

I.96 

1.96 

i  on  2 

0.00 

i  on  i 

2-33 

2-34 

2.19 

2.  II 

2.06 

2.03 

i  on  2 

O.  IO 

i  on  2 

1-73 

I  .90 

1.99 

2.02 

2.02 

2.01 

i  on  2 

O.  2O 

i  on  5 

1.82 

1.97 

1.94 

1-93 

i-95 

1.97 

Vertical 

0.80 

Vertical 

1-43 

1.47 

1-57 

1.66 

1.77 

1.87 

Vertical 

0.80* 

Vertical 

1.63 

1.66 

1.70 

1.79 

1.87 

1.92 

Vertical 

2.00 

Vericalt 

1-38 

1-43 

1-37 

i.39 

1.44 

1.49 

Vertical 

2.OO* 

Vertica 

1.50 

1-56 

1-57 

1.58 

i.  60 

1.63 

i  on  i 

Round 

Vertica 

1.63 

1-75 

1.91 

1.96 

1.99 

2.OI 

Explanation  in  Art.  72. 


558 


HYDRAULIC  TABLES 


TABLE  31.     COEFFICIENTS  FOR  CONICAL  TUBES 

For  all  Measures 


Angle  of  Cone 

Discharge 
c 

Velocity 
c\ 

Contraction 
c' 

0°       00' 

0.829 

0.829 

I  .OO 

I    36 

0.866 

0.867 

4      10 

0.912 

0.910 

7      52 

0.930 

0.932 

0.998 

10         20 

0.938 

0.951 

0.986 

13         24 

0.946 

0.963 

0.983 

16      36 

0.938 

0.971 

0.966 

21         00 

0.919 

0.972 

0-945 

29      58 

0.895 

0.975 

0.918 

48     50 

0.847 

0.984 

0.861 

Explanation  in  Art.  77. 


TABLE  32.     VERTICAL  JETS  FROM  SMOOTH  NOZZLES 

English  Measures 


Indicated 

From  $-inch  Nozzle 

From  i  -inch  Nozzle 

From  i^-inch  Nozzle 

Pressure  at 

Entrance 

to  Nozzle 
Pounds  per 

Height  in  Feet 

Dis- 
charge 

Height  in  Feet 

Dis- 
charge 

Height  in  Feet 

Dis- 
charge 

Gallons 

Gallons 

Gallons 

Inch 

A 

B 

per 

Minute 

A 

B 

per 
Minute 

A 

B 

per 

Minute 

10 

20 

17 

52 

21 

18 

93 

22 

19 

148 

20 

40 

33 

73 

43 

35 

132 

44 

37 

209 

30 

59 

48 

90 

63 

5i 

161 

66 

53 

256 

40 

78 

60 

104 

83 

64 

1  86 

86 

67 

.296 

50 

93 

67 

116 

IOI 

73 

208 

107 

77 

331 

60 

104 

72 

127 

117 

79 

228 

126 

85 

363 

70 

114 

76 

137 

130 

85 

246 

140 

9i 

392 

80 

123 

79 

147 

140 

89 

263 

150 

95 

419 

90 

129 

81 

156 

147 

92 

279 

157 

99 

444 

IOO 

134 

83 

164 

152 

96 

295 

161 

IOI 

468 

Explanation  in  Art. 


HYDRAULIC  TABLES 


559 


TABLE  33.     FRICTION  FACTORS  FOR    CLEAN  IRON  PIPES 

Arguments  in  English  Measures 


Diameter 
in 
Feet 

Velocity  in  Feet  per  Second 

i 

2 

3 

4          1 

6 

10 

IS 

0.05 

0.047 

O.O4I 

0.037 

0.034 

0.031 

O.O29 

0.028 

O.  I 

.038 

.032 

.030 

.028 

.026 

.024 

.023 

0.25 

0-5 

.032 
.028 

.028 
.026 

.026 
.025 

.025 

.023 

.024 
^ 
.022 

.022 
t  V»  J 
.O2O 

.021 
.019 

0.75 

.026 

.025 

.024 

.022 

.021 

.OI9 

.018 

i. 

.025 

.024 

.023 

.022 

.O2O 

.018 

.017 

1-25 

.024 

.023 

.022 

.021 

.019 

.017 

.Ol6 

i-5 

.023 

.022 

.O2I 

.O2O 

.018 

.Ol6 

.015 

1.75 

.022 

.021 

.020 

.018 

.017 

.015 

.014 

2. 

.O2I 

.O2O 

.OI9 

.017 

.Ol6 

.014 

.013 

2-5 

.020 

.019 

.018 

.016 

•015 

.013 

.012 

3- 

.019 

.018 

.Ol6 

.015 

.014 

.013 

.012 

3-5 

.018    ' 

.017 

.016 

.OI4 

.013 

.012 

4- 

.017 

.Ol6 

•015 

.013 

.OI2 

.Oil 

5- 

.016 

.015 

.014 

.013 

.012 

6. 

.015 

.014 

.013 

.OI2 

.Oil 

Explanation  in  Art.  86. 


TABLE  34.     FRICTION  FACTORS  FOR  CLEAN  IRON  PIPES 
Arguments  in  Metric  Measures 


Diameter  in 
Centimeters 

Velocity  in  Meters  per  Second 

0.3 

0.6 

I.O 

i-S 

2-5 

4-5 

i-5 

0.047 

0.041 

0.036 

0.033 

0.030 

0.028 

3- 

.038 

.032 

.030 

.027 

.025 

.023 

8. 

.031 

.028 

.026 

.024 

.023 

.O2I 

16. 

.027 

.026 

.025 

.023 

.O2I 

.019 

30. 

.025 

.024 

•  023 

.O2I 

.019 

.017 

40. 

.024 

.023 

.022 

.019 

.018 

.Ol6 

60. 

.022 

.020 

.OI9 

.017 

.015 

.013 

90. 

.OI9 

.018 

.Ol6 

.015 

.013 

.OI2 

120. 

.017 

.016 

.015 

•  013 

.OI2 

180. 

.015 

.014 

.013 

.OI2 

Explanation  in  Art. 


560 


HYDRAULIC  TABLES 


TABLE  35.     FRICTION  HEAD  FOR  100  FEET  OF  CLEAN  IRON  PIPE 

English  Measures 


Diameter 
in  Feet 

Velocity  in  Feet  per  Second 

i 

2 

3 

4 

6 

10 

15 

Feet 

Feet 

Feet 

Feet 

Feet 

Feet 

Feet 

0.05 

1.46 

5-10 

10.3 

16.9 

34-7 

O.  I 

0-59 

1.99 

4.20 

6.97 

!4-5 

37-3 

0.25 

.20 

0.70 

1  .46 

2.40 

5-37 

13-7 

29.4 

0-5 

.09 

0.32 

0.70 

I.I4 

2.46 

6.22 

!3-3 

0-75 

•05 

.21 

0.45 

0.73 

1-57 

3-94 

8.40 

i  . 

.04 

•15 

•32 

•  55 

I  .  12 

2.80 

5-95 

1-25 

•03 

.  II 

•25 

.42 

0.85 

2.  II 

4.48 

i-5 

.02 

.09 

.20 

•  33 

•67 

1.66 

3.50 

i-75 

.02 

.07 

.16 

.26 

•54 

1-33 

2.80 

2. 

.02 

.06 

•13 

.21 

•  45 

1.09 

2.27 

2-5 

.OI 

.05 

.10 

.16 

•34 

0.81 

1.68 

3- 

.OI 

.04 

.07 

.  12 

.26 

•67 

i  .40 

3-5 

.01 

•03 

.06 

.  10 

.21 

•  53 

4- 

.02 

•05 

.08 

•17 

.42 

5- 

.02 

.04 

.06 

•13 

6. 

.01 

•03 

•05 

.10 

Explanation  in  Art. 


TABLE  36.    FRICTION  HEAD  FOR  100  METERS  OF  CLEAN  IRON  PIPE 

Metric  Measures 


Diameter  in 
Centimeters 

Velocity  in  Meters  per  Second 

0.3 

0.6 

10 

i-5 

2-5 

45 

Meters 

Meters 

Meters 

Meters 

Meters 

Meters 

1.5 

1.44 

5-02 

12.2 

3- 

0.58 

I.96 

5.10 

10.3 

26.6 

8. 

0.18 

0.64 

1.66 

3-45 

9-23 

27.1 

16. 

.08 

•30 

.80 

1.65 

4.09 

12.3 

30. 

.04 

•15 

•39 

0.80 

2.  02 

5-85 

40. 

•03 

.  IO 

.28 

•54 

1-43 

4-13 

60. 

.02 

.06 

.16 

•33 

0.80 

2.24 

90. 

.OI 

.04 

.09 

•19 

.46 

1.38 

1  20. 

.02 

.06 

.  12 

•32 

180. 

.OI 

.04 

.08 

Explanation  in  Art.   104. 


HYDRAULIC  TABLES 


561 


TABLE  37.     COEFFICIENTS  FOR  CIRCULAR  CONDUITS 

English  Measures 


Diameter 
in  Feet 

Velocity  in  Feet  per  Second 

i 

2 

3 

4 

6 

IO 

IS 

I. 

96 

IO4 

109 

112 

116 

121 

124 

i.5 

103 

III 

116 

119 

123 

129 

I32 

2. 

109 

116 

121 

I24 

129 

134 

138 

2.5 

H3 

1  20 

125 

128 

133 

139 

143 

3. 

117 

124 

128 

132 

136 

H3 

J47 

3-5 

120 

127 

131 

135 

139 

I46 

151 

4- 

123 

130 

134 

137 

142 

150 

155 

5. 

128 

134 

139 

142 

147 

155 

6. 

132 

138 

142 

145 

150 

7- 

135 

141 

145 

149 

153 

8. 

137 

143 

I48 

151 

Explanation  in  Art.  107. 


TABLE  38.     COEFFICIENTS  FOR  CIRCULAR  CONDUITS 

Metric  Measures 


Velocity  in  Meters  per  Second 

Diameter 

0.3 

0.6 

O.Q 

1.5 

3.0 

4-5 

0.3 

53 

57 

60 

63 

67 

68 

0.5 

57 

61 

64 

67 

7i 

73 

0.7 

61 

65 

68 

7i 

76 

78 

0.9 

64 

68 

70 

74 

79 

81 

I.I 

66 

70 

72 

76 

81 

83 

1.3 

68 

72 

74 

78 

83 

1.6 

72 

74 

77 

80 

2.O 

74 

77 

79 

83 

2.4 

76 

79 

82 

Explanation  in  Art.  119. 


562 


HYDRAULIC  TABLES 


TABLE  39.     CROSS-SECTIONS  OF  CIRCULAR  CONDUITS 
For  all  Measures 


Depth 
a 

Wetted 
Perimeter 

P 

Sectional 
Area 
a 

Hydraulic 
Radius 

r 

Velocity 
\fi- 

Discharge 
a  \/r~ 

Full         .1.0 

3.142 

0.7854 

0.25 

0.5 

0.393 

o.95 

2.691 

0.7708 

0.286 

0.535 

.413 

0.9 

2.498 

0.7445 

0.298 

0.546 

.406 

0.81 

2.240 

0.6815 

0.3043 

0.552 

.376 

0.8 

2.214 

0.6735 

0.3042 

0.552 

.372 

0.7 

1.983 

0.5874 

0.296 

0.544 

.320 

0.6 

1.772 

0.4920 

0.278 

0.527 

•259 

Half  Full  0.5 

I.57I 

0.3927 

0.25 

0.5 

.I96 

0.4 

1.369 

0.2934 

0.214 

0.463 

.136 

0.3 

I.I59 

o.  1981 

o.  171 

0.414 

.0820 

0.2 

0.927 

0.1118 

O.  121 

0.348 

.0389 

O.I 

0.643 

o  .  0408 

0.0635 

0.252 

.0103 

Empty     o.o 

o.o. 

o.o 

0.0 

o.o 

0.0 

Explanation  in  Art.  108. 


TABLE  40.     COEFFICIENTS  FOR  RECTANGULAR  CONDUITS 

English  Measures 


Unplaned  Plank 
&=3-93  Feet 

Unplaned  Plank 
fc  =  6.53  Feet 

Neat  Cement 
&  =  5-94  Feet 

Brick 
6=6.27  Feet 

d 

c 

d 

c 

d 

c 

d 

c 

0.27 

99 

O.2O 

89 

0.18 

116 

0.20 

89 

•41 

108 

•30 

101 

.28 

125 

.31 

98 

.67 

112 

.46 

109 

•43 

132 

•49 

104 

.89 

114 

.60 

H3 

•56 

135 

•  57 

105 

I  .OO 

114 

.72 

116 

.63 

136 

.65 

105 

I.I9 

116 

.78 

116 

.69 

136 

.71 

1  06 

1.29 

117 

.89 

1x8 

.80 

137 

.85 

107 

1.46 

118 

•  94 

120 

•91 

138 

•  97 

no 

Explanation  in  Art.   109. 


HYDRAULIC  TABLES 


563 


TABLE  41.     COEFFICIENTS  FOR  RECTANGULAR  CONDUITS 

Metric  Measures 


Unplaned  Plank 

Unplaned  Plank 

Neat  Cement 

Brick 

6=1.2  Meters 

6  =  2  Meters 

6=1.8  Meters 

6  —  i  .9  Meters 

d 

c 

d 

c 

d 

C 

d 

C 

0.08 

55 

O.o6 

49 

O.O6 

64 

0.06 

49 

.15 

60 

.09 

56 

.08 

69 

.09 

54 

.18 

61 

•13 

60 

•13 

73 

•15 

57 

.27 

63 

.18 

62 

•17 

74 

•17 

58 

•30 

63 

.20 

64 

•19 

75 

.20 

58 

.36 

64 

•24 

64 

.21 

75 

.22 

59 

•39 

65 

.27 

65 

.24 

76 

.26 

60 

•44 

65 

.29 

66 

.27 

76 

•30 

61 

Explanation  in  Art.  119- 


TABLE  42.     KUTTER'S  COEFFICIENTS  FOR  SEWERS 
English  Measures 


Hydraulic 

5  =  0  00005 

5=0  0001 

5  =  0  01 

Radius  r 

in  Feet 

n  =  o.oi5 

«  =  o  017 

n  =  o  015 

n  =  o  017 

n«=o.oi5 

n  =  o.oi7 

0.2 

52 

43 

58 

48 

68 

57 

0-3 

60 

5i 

66 

56 

76 

64 

0.4 

0.6 

65 
76 

56 
65 

73 
82 

61 
70 

83 
90 

,70 
^76 

0.8 

82 

72 

87 

76 

95 

82 

i. 

88 

77 

92 

80 

99 

87 

1-5 

100 

86 

103 

89 

1  08 

93 

2. 

1  06 

94 

108 

96 

in 

99 

3. 

116 

103 

118 

104 

118 

105 

Explanation  in  Art.  112. 


564 


HYDRAULIC  TABLES 


TABLE  43.     KUTTER'S  COEFFICIENTS  FOR  SEWERS 

Metric  Measures 


Hydraulic 
Radius  r 
in  Meters 

5  =  0.00005 

5=0.0001 

5  =  0.01 

«  =  o.ois 

n  =  o.oi7 

«  =  o.ois 

«  =  o.oi7 

w  =  o.ois 

n  =  o.oi7 

0.05 

26 

22 

31 

25 

37 

30 

O.I 

34 

29 

37 

32 

43 

36 

0.15 

39 

33 

42 

36 

48 

40 

0.2 

43 

38 

46 

40 

5i 

43 

0-3 

49 

42 

51 

44 

55 

48 

0.5 

56 

48 

57 

50 

60 

52 

0.7 

62 

54 

62 

55 

63 

56 

1.0 

67 

59 

67 

58 

66 

59 

Explanation  in  Art.  119. 


TABLE  44.     KUTTER'S  COEFFICIENTS  FOR  CHANNELS 

English  Measures 


Hydraulic 
Radius  r 
in  Feet 

5=0.00005 

5  =  O.OOOI 

5=0.01 

n  =  o.025 

n  =  o.o30 

«=0.025 

w  =  0.030 

n  —  0.025 

«  =  0.030 

0-5 

38 

31 

41 

33 

47 

37 

I  . 

49 

40 

52 

42 

56 

45 

1-5 

57 

47 

59 

48 

62 

5i 

2. 

64 

52 

65 

53 

67 

54 

3- 

72 

59 

72 

59 

72 

60 

4- 

77 

64 

77 

64 

76 

63 

5. 

81 

68 

80 

68 

79 

66 

6. 

86 

72 

84 

7i 

80 

68 

8. 

9i 

76 

87 

74 

82 

70 

10. 

96 

80 

91 

80 

85 

73 

IS- 

105 

89 

97 

84 

90 

77 

25- 

114 

100 

IOI 

92 

95 

82 

Explanation  in  Art.  113. 


HYDRAULIC  TABLES 


565 


TABLE  45.     KUTTER'S  COEFFICIENTS  FOR  CHANNELS 
Metric  Measures 


Hydraulic 

5=0.00005 

5  =  0.0001 

5  =  0.01 

Radius  r 

in  Meters 

n  =  o.o25 

tt-0.030 

«  =  0.025 

n-o.030 

n  =  0.025 

w  =  0.030 

0.2 

22 

18 

24 

19 

27 

21 

0-3 

27 

22 

29 

33 

31 

25 

0.5 

32 

27 

34 

27 

35 

28 

0.7 

36 

30 

37 

30 

38 

31 

1.0 

40 

33 

40 

33 

40 

33 

1.5 

45 

38 

44 

38 

43 

36 

2. 

48 

41 

47 

40 

45 

38 

3- 

53 

44 

50 

44 

47 

40 

5- 

59 

50 

53 

47 

51 

43 

Explanation  in  Art.  115. 

TABLE  46.     BAZIN'S  COEFFICIENTS  FOR  CHANNELS 

English  Measures 


Hydraulic 

Radius  r 

w  =  o.o6 

ra  =  o.i6 

w  =  o.46 

w  =  o.8S 

m=  1.30 

w=i.75 

in  Feet 

I  . 

142 

122 

86 

62 

47 

38      , 

2. 

146 

131 

IOO 

76 

60 

49 

3- 

I48 

135 

107 

84 

67 

56 

5.- 

150 

140 

115 

94 

78 

67 

7- 

151 

142 

120 

IOO 

84 

75 

10. 

152 

144 

125 

106 

9i 

79 

Explanation  in  Art.  115. 

TABLE  47.     BAZIN'S  COEFFICIENTS  FOR  CHANNELS 

Metric  Measures 


Hydraulic 
Radius  r 
in  Meters 

m  =  0.06 

m  =  o.i6 

w  =  o.46 

w  =  o.8s 

m=  1.30 

w=i.75 

0-5 

80.2 

70.9 

52.7 

39-5 

30.6 

25-0 

I. 

82.1 

75-0 

6o.O 

47-0 

37-8 

31-3 

2. 

83.4 

78.1 

65.6 

54-3 

45-3 

38.9 

3- 

84.0 

79-6 

68.7 

58.3 

49-7 

43-3 

5- 

84.7 

81.2 

72.1 

63.0 

55-0 

48.8 

Explanation  in  Art.  119. 


566 


HYDRAULIC  TABLES 


TABLE  48.     VALUES  OF  THE  BACKWATER  FUNCTION 


D 

D 

D 

ld\ 

D 

ld\ 

d 

*w 

d 

*y 

d 

Hw 

d 

*w 

i. 

CO 

0.954 

0.9073 

0.845 

0.5037 

0.61 

0.2058 

0.999 

2.1834 

•  952 

.8931 

.840 

.4932 

.60 

.1980 

.998 

.9523 

•  950 

.8795 

•  835 

.4831 

•59 

.1905 

•  997 

.8172 

.948 

.8665 

.830 

•4733 

•58 

.1832 

.996 

.7213 

.946 

.8539 

.825 

.4637 

•  57 

.1761 

•  995 

.6469 

•  944 

.8418 

.820 

•  4544 

•  56 

.  1692 

•994 

.5861 

•942 

.8301 

•8i5 

•4454 

•  55 

.1625 

•993 

•  5348 

.940 

.8188 

.810 

.4367 

•54 

•  1560 

.992 

.4902 

.938 

.8079 

•  805 

.4281 

•53 

.1497 

.991 

•4510 

.936 

.7973 

.800 

.4198 

•52 

.1435 

.990 

.4159 

•  934 

.7871 

•  795 

.4117 

.51 

.1376 

.989 

•3841 

•  932 

.7772 

•  790 

.4039 

•  50 

.1318 

.988 

•3551 

•  930 

.7675 

•  785 

.3962 

•49 

.1262 

•987 

.3284 

.928 

.7581 

.780 

.3886 

•48 

.  1207 

.986 

.3037 

.926 

.7490 

•  775 

'  .3813 

•47 

.1154 

•985 

.2807 

.924 

.7401 

•  770 

•3741 

.46 

.  IIO2 

.984 

•2592 

.922 

.7315 

.765 

.3671 

•45 

.1052 

•983 

.2390 

.920 

.7231 

.760 

.3603 

•44 

.1003 

.982 

.2199 

.918 

.7149 

•  755 

•3536 

•43 

.0995 

.981 

.2019 

.916 

.7069 

•  750 

•3470 

.42 

.0909 

.980 

.1848 

.914 

.6990 

•  745 

.3406 

.41 

.0865 

•979 

.1686 

.912 

.6914 

•  740 

•  3343 

•  40 

.0821 

•978 

.1531 

.910 

.6839 

•735 

.3282 

•39 

.0/79 

•977 

•1383 

.908 

.6766 

-730 

•  3221 

•38 

.0738 

.976 

.1241 

.906 

.6695 

•  725 

.3162 

•37 

.0699 

•  975 

.1105 

•904 

.6625 

.720 

•3104 

•36 

.0660 

•974 

.0974 

.902 

.6556 

.715 

•3047 

•35 

.0623 

•  973 

.0848 

.900 

.6489 

.710 

.2991 

•34 

.0587 

.972 

•0727 

.895 

.6327 

•  705 

•2937 

•33 

•0553 

.971 

.0610 

.890 

.6173 

.70 

.2883 

•32 

.0519 

•  970 

.0497 

•  885 

.6025 

.69 

•2778 

•  30 

.0455 

.968 

.0282 

.880 

.5884 

.68 

.2677 

.28 

•0395 

.966 

.0080 

•  875 

•  5749 

.67 

•  2580 

•  25 

.0314 

.964 

o  .  9890 

.870 

•  5619 

.66 

.2486 

.20 

.O2OI 

.962 

.9709 

.865 

•5494 

•65 

•2395 

•  15 

.0113 

.960 

•  9539 

.860 

•5374 

.64 

.2306 

.  10 

.0050 

•  958 

.9376 

.855 

•  5258 

•63 

.2221 

•  05 

.0015 

.956 

.9221 

.850 

•  5146 

.62 

.2138 

.00 

.0000 

Explanation  in  Art.   131. 


HYDRAULIC  TABLES 


567 


TABLE  49.     VALUES  OF  THE  DROP-DOWN  FUNCTION 


d 

,  /  d\ 

d 

^id\ 

d 

.fd\ 

d 

fd\ 

D 

*w 

D 

*(D) 

D 

*(D) 

D 

*(P) 

I  . 

00 

0.954 

0.8916 

0.845 

0.4478 

0.61 

0.0454 

0.999 

2.I83I 

.952 

.8767 

.840 

•4353 

.60 

•0325 

.998 

.9517 

.950 

.8624 

.835 

.4232 

•59 

.0199 

•  997 

.8162 

.948 

.8487 

.830 

.4114 

.58 

+  0.0074 

.996 

.7206 

.946 

.8354 

.825 

.3988 

•  57 

—  0.0050 

•  995 

.6452 

•  944 

.9226 

.820 

.38*6 

•  56 

—  .0172 

•  994 

.5841 

.942 

.8102 

•815 

.3776 

•55 

-  .0293 

•  993 

.5324 

.940 

.7982 

.810 

.3668 

•54 

—  .0412 

•  992 

.4876 

.938 

.7866 

.805 

.3562 

•53 

~  .0530 

.991 

.4486 

.936 

•  7753 

.800 

•3459 

•  52 

-  .0647 

.990 

.4125 

•934 

.7643 

•  795 

-3357 

•  5i 

-  .0763 

.989 

.3804 

•  932 

•  7537 

•  790 

.3258 

•  50 

—  .0878 

.988 

•3511 

•  930 

•  7433 

.785 

.3160 

•49 

-  .0991 

.987- 

.3241 

.928 

•7332 

.780 

.3064 

•  48 

—  .  1  104 

.986 

.2990 

.926 

.7234 

•  775 

.2970 

•47 

—  .1216 

.985 

•2757 

•  924 

•  7138 

.770 

•  2877 

•46 

-  .1327 

.984 

.2538 

.922 

.7045 

-765 

•  2785 

•45 

-  .1438 

•983 

.2323 

.920 

•6953 

.760 

.2696 

•44 

-  -1547 

.982 

.2139 

.918 

.6864 

•  755 

.2607 

•43 

-  -1656 

.981 

•1955 

.916 

.6776 

•  750 

.2520 

•42 

-  -1765 

.980 

.1781 

.914 

.6691 

•  745 

•2434 

•4i 

-  .1872 

•  979 

.  l6l5 

.912 

.6607 

•  740  ' 

•2350 

.40 

-  .1980 

.978 

•1457 

.910 

•6525 

•735 

.2260 

•39 

—  .  2086 

•  977 

.1305 

.908 

.6445 

•  730 

.2184 

•38 

—  .2192 

.976 

.  1160 

.906 

.6366 

•725 

.2102 

•  37 

—  .2298 

-975 

.  1020 

.904 

.6289 

.720 

.2022 

•  36 

-  -2403 

•974 

.0886 

.902 

.6213 

.715 

•1943 

•35 

-  .2508 

•973 

.0757 

.900 

.6138 

.710 

.1864 

•34 

—  .2612 

.972 

.0632 

.895 

•5958 

.705 

.1787 

•33 

—  .2716 

•  97J 

.0512 

.890 

.5785 

.70 

.1711 

•32 

-  .2819 

•  970 

.0396 

.885 

.5619 

.69 

.1560 

•30 

-  -3025 

.968 

.0174 

.880 

•5459 

.68 

•1413 

.28 

-  .3230 

.966 

0.9965 

.875 

.5305 

.67 

.1268 

•25 

-  .3536 

.964 

.9767 

.870 

•5156 

.66 

.1127 

.20 

-  .4042 

.962 

.9580 

.865 

.5012 

•65 

.0987 

•15 

-  -4544 

.960 

.9402 

.860 

.4872 

.64 

.0851 

.10 

-  .  5046 

.958 

.9233 

•  855 

•  4737 

•63 

.0716 

•05 

-  .5546 

.956 

.9071 

.850 

.4605 

.62 

.0584 

.00 

—  .  6046 

Explanation  in  Art.   132. 


568 


MATHEMATICAL  TABLES 


TABLE  50.     SQUARES  OF  NUMBERS 


n 

01234 

56789 

Diff. 

I.O 

I.  ooo  1.020  1.040  1.061   1.082 

1.103  1-124  I.I45   1.166  1.188 

22 

i.i 

1.  210    1.232    1.254    1-277    1-300 

1.323  1.346  1.369  1.392  1.416 

24 

1.2 

1.440    1.464    1.488    I.5I3    1.538 

1.563  1.588  1.613  1.638  1.664 

26 

i-3 

1.690    I.7I6    1.742    1.769    1.796 

1.823  1.850  1.877  1-904  1.932 

28 

1.4 

1.960    1.988    2.016    2.045    2.074 

2.103    2.132    2.l6l    2.190    2.220 

30 

1-5 

2.250    2.280    2.310    2.341    2.372 

-2.403    2.434    2.465    2.496    2.528 

32 

1.6 

2.560    2.592    2.624    2.657    2.690 

2.723    2.756    2.789    2.822    2.856 

34 

i-7 

2.890    2.924    2.958    2.993    3.028 

3.063    3.098    3.133    3.168    3.204 

36 

1.8 

3.240    3.276    3.312    3.349    3.386 

3.423    3.460    3.497    3.534    3.572 

38 

1.9 

>3.6io  3.648  3.686  3.725  3.764 

3.803    3.842    3.88l    3.920    3.960 

40 

2.0 

4.000  4.040  4.080  4.121  4.162 

4.203    4.244    4.285    4.326    4.368 

42 

2.1 

4.410  4.452  4.494  4.537  4.580 

4.623    4.666    4.709    4.752    4.796 

44 

2'.  2 

4.840  4.884  4.928  4.973  5.018 

5.063    5.108    5-153    5.198    5-244 

46 

2-3 

5.290  5.336  5.382  5.429  5.476 

5.523    5.570    5.617    5.664    5.712 

48 

2.4 

5.760  5.808  5.856  5-905  5-954 

6.003  6.052  6.101  6.150  6.200 

50 

2-5 

6.250  6.300  6.350  6.401  6.452 

6.503  6.554  6.605  6.656  6.708 

52 

2.6 

6.760  6.812  6.864  6.917  6.970 

7.023  7.076  7.129  7.182  7.236 

54 

2.7 

7.290  7.344  7.398  7.453  7.508 

7.563  7.618  7.673  7.728  7.784 

56 

2.8 

7.840  7.896  7.952  8.009  8-066 

8.123  8.180  8.237  8.294  8.352 

58 

2.9 

8.410  8.468  8.526  8.585  8.644 

8.703  8.762  8.821  8.880  8.940 

60 

3-o 

9.000  9.060  9.120  9.181  9.242 

9-303  9-364  9-425  9.486  9.548 

62 

3-i 

9.610  9.672  9.734  9.797  9.860 

9.923  9.986  10.05  io.ii   10.18 

6 

3-2 

10.24  10.30  10.37  10.43  10.50 

10.56  10.63  10.69  10.76  10.82 

7 

3-3 

10.89  10.96  n.  02  11.09  ii.  16 

11.22  11.29  11-36  11.42  11.49 

7 

3-4 

11.56  11.63  11.70  11.76  11.83 

11.90    11-97    12.04    12.  II    12.  18 

7 

3-5 

12.25  12.32  12.39  12.46  12.53 

12.  60    12.67    12.74    12.82    12.89 

7 

3-6 

12.96  13.63  13.10  13.18  13.25 

13.32    13.40    13.47    13.54    I3.62 

7 

3-7 

13.69  13.76  13.84  13.91  13.99 

14.06    14.14    14.21    1429    14.36 

8 

3-8 

14.44  14-52  14-59  14-67  14.75 

14.82    14.90    14.98    15.05    15.13 

8 

3-9 

15.21   15.29  15.37  15.44  15.52 

15.60    15.68    15.76    15.84    15.92 

8 

4.0 

16.00  16.08  16.16  16.24  16.32 

16.40    16.48    16.56    16.65    16.73 

8 

4.1 

16.81  16.89  l6-97  17-06  17.14 

17.22    17.31    17.39    17-47    I7-56 

8 

4.2 

17.64  17.72  17.81  17.89  17.98 

18.06  18.15  18.23  18.32  18.40 

9 

4-3 

18.49  18.58  18.66  18.75  18.84 

18.92  19.01  IQ.IO  19.18  19.27 

9 

4.4 

19.36  19.45  19.54  19.62  19.71 

19.80  19.89  19.98  20.07  20.16 

9 

4-5 

20.25  20.34  20.43  20.52  20.61 

20.70  20.79  20.88  20.98  21.07 

9 

4.6 

21.  16  21.25  21.34  21.44  21.53 

21.62    21.72    21.  8l    21.90    22.00 

9 

4-7 

22.09  22.18  22.28  22.37  22.47 

22.56  22.66  22.75  22.85  22.94 

10 

4.8 

23.04  23.14  23.23  23.33  23.43 

23.52  23.62  23.72  23.81  23.91 

10 

4.9 

24.01  24.11  2421  24.30  24.40 

24.50  24.60  24.70  24.80  24.90 

10 

5-0 

25.00  25.10  25.20  25.30  25.40 

25.50  25.60  25.70  25.81  25.91 

10 

5-i 

26.01  26.11  26.21  26.32  26.42 

26.52  26.63  26.73  26.83  26.94 

10 

5-2 

27.04  27.14  27.25  27.35  27.46 

27.56  27.67  27.77  27.88  27.98 

ii 

5-3 

28.09  28.20  28.30  28.41  28.52 

28.62  28.73  28.84  28.94  29.05 

ii 

5-4 

29.16  29.27  29.38  29.48  29.59 

29.70  29.81  29.92  30.03  30.14 

ii 

72 

01234 

56789 

Diff. 

Explanation  in  Art.  196. 


MATHEMATICAL  TABLES 


TABLE  50.     SQUARES  OF  NUMBERS 


n 

01234 

56789 

Diff. 

5-5 

30.25  30.36  30.47  30.58  30.69 

30.80  30.91  31.02  31.14  31.25 

n 

5-6 

31.36  31.47  31.58  31-70  31.81 

31.92  32.04  32.15  32.26  32.38 

ii 

5-7 

32.49  32.60  32.72  32.83  32.95 

33.06  33.18  33.29  33.41  33.52 

12 

5-8 

33.64  33  76  33.87  33.99  34.11 

34.22  34.3.1   34.46  34.57  34.69 

12 

5-9 

34.81  34.93  35-05  35.i6  35.28 

35-40  35-52  35.64  35.76  35.88 

12 

6.0 

36.00  36.12  36.24  36.36  36.48 

36.60  36.72  36.84  36.97  37.09 

12 

6.1 

37-21  37.33  37-45  37.58  37.70 

37.82  37.95  38.07  38.19  38.32 

12 

6.2 

38.44  38.56  38.69  38.81  38.94 

39.06  39.19  39.31  39.44  39.56 

13 

6-3 

39.69  39.82  39.94  40.07  40.20 

40.32  40.45  40.58  40.70  40.83 

13 

6.4 

40.96  41.09  41.22  41.34  41.47 

41.60  41.73  41.86  41.99  42.12 

13 

6-5 

42.25  42.38  42.51  42.64  42.77 

42.90  43.03  43.16  43.30  43.43 

13 

6.6 

43.56  43.69  43.82  43.96  44.09 

44.22  44.36  44.49  44.62  44.76 

13 

6.7 

44.89  45.02  45.16  45.29  45.43 

45.56  45.70  45.83  45.97  46.10 

14 

6.8 

46.24  46.38  46.51  46.65  46.79 

46.92  47.06  47.20  47.33  47.47 

14 

6-9 

47.61  47.75  47-89  48.02  48.16 

48.30  48.44  48.58  48.72  48.86 

14 

7.0 

49.00  49.14  49-28  49.42  49-56 

49.70  49.84  49.98  50.13  50.27 

14 

7-1 

50.41  50.55   50.69  50.84  50.98 

51.12  51.27  51.41  51.55  51.70 

14 

7.2 

51.84  51.98  52.13  52.27  52-42 

52.56  52.71  52.85  53.00  53.14 

15 

7-3 

53-29  53-44  53.58  53-73  53-88 

54.02  54.17  54-32  54.46  54.61 

15 

7-4 

54.76  54.91  55-o6  55.20  55.35 

55-50  55.65  55.80  55.95  56.10 

15 

7-5 

56.25  56.40  56.55  56.70  56.85 

57.00  57.15  57.30  57.46  57-61 

15 

7-6 

57.76  57-91  58.o6  58.22  58.37 

58.52  58.68  58.83  58.98  59.14 

15 

7-7 

59.29  59.44  59-6o  59-75  59.91 

60.06  60.22  60.37  60.53  60.68 

16 

7-8 

60.84  61.00  61.15  61.31  61.47 

61.62  61.78  61.94  62.09  62.25 

16 

7-9 

62.41  62.57  62.73  62.88  63.04 

63.20  63.36  63.52  63.68  63.84 

16 

8.0 

64.00  64.16  64.32  64.48  64.64 

64.80  64.96  65.12  65.29  65.45 

16 

8.1 

65.61  65.77  65.93  66.10  66.26 

66.42  66.59  66.75  66.91  67.08 

16 

8.2 

67.24  67.40  67.57  67.73  67.90 

68.06  68.23  68.39  68.56  68.72 

17 

8-3 

68.89  69.06  69.22  69.39  69.56 

69.72  69.89  70.06  70.22  70.39 

17 

8.4 

70.56  70.73  70.90  71.06  71.23 

71.40  71-57  71-74  7I-9I  7208 

17 

8-5 

72.25  72.42  72-59  72.76  72.93 

73.10  73.27  73.44  73.62  73.79 

17 

8.6 

73.96  74-13  74.30  74.48  74.65 

74.82  75.oo  75.17  75-34  75.52 

17 

8-7 

75.69  75.86  76.04  76.21  76.39 

76.56  76.74  76.91  77.09  77.26 

18 

8.8 

77.44  77-62  77.79  77-97  78.15 

78.32  78.50  78.68  78.85  79.03 

18 

8.Q 

79.21  79.39  79-57  79-74  79-92 

80.10  80.28  80.46  80.64  80.82 

18 

9.0 

81.00  81.  18  81.36  81.54  81.72 

81.90  82.08  82.26  82.45  82.63 

18 

9.1 

82.81  82.99  83.17  83.36  83.54 

83.72  83.91  84.09  84.27  84.46 

18 

9.2 

84.64  84.82  85.01  85.19  85.38 

85.56  85.75  85.93  86.12  86.30 

19 

9-3 

86.49  86.68  86.86  87.05  87.24 

87.42  87.61  87.80  87.98  88.17 

19 

9-4 

88.36  88.55  88.74  88.92  89.11 

89.30  89.49  89.68  89.87  90.06 

19 

9-5 

90.25  90.44  90.63  90.82  91.01 

91.20  91.39  91.58  91-78  91.97 

19 

9.6 

92.16  92.35  92.54  92.74  92.93 

93.12  93.32  93.51  93.70  93.90 

19 

9-7 

94.09  94.28  94.48  94.67  94.87 

95.06  95.26  95.45  95.65  95.84 

20 

9.8 

96.04  96.24  96.43  96.63  96.83 

97.02  97.22  97.42  97.61  97.81 

20 

9-9 

98.01  98.21  98.41  98.60  98.80 

99.00  99.20  99.40  99.60  99.80 

20 

n 

01234 

56789 

Diff. 

Explanation  in  Art.  19- 


570 


MATHEMATICAL  TABLES 
TABLE  51.     AREAS  OF  CIRCLES 


d 

01234 

56789 

DifL 

I.O 

.7854   .8012    .8171    .8332    .8495 

.8659     .8825      .8992    .9161     .9331 

i.i 

.9503     .9677     .9852     1.003     I.  021 

1.039     1.057     1-075    1-094    I.  112 

1.2 

I.I3I     I.I50     1.169     I.I88     1.  208 

1.227    L247    1.267  1.287  J-S0? 

19 

1-3 

1.327     1.348     1.368     1.389     I.4IO 

1.431  1.453  1.474  1.496  1.517 

21 

1-4 

1.539     I'S^I     1.584     1.  606     1.629 

1.651   1.674  1-697  1.720  1.744 

22 

1.5 

1.767     I.79I     I.8I5     1.839     1-863 

1.887    1.911    1.936  1.961  1.986 

24 

1.6 

2.  Oil     2.036     2.061     2.087     2.  112 

2.138    2.164    2.190  2.217  2.243 

26 

i-7 

2.270    2.297     2.324    2.351     2.378 

2.405  2.433  2.461  2.488  2.516 

27 

1.8 

2.545     2.573     2.602     2.630    2.659 

2.688    2.717    2.746  2.776  2.806 

29 

1.9 

2.835     2.865     2.895     2.926     2.956 

2.986   3.017    3.048  3.079  3.110 

30 

2.O 

3.142     3.173     3.205     3.237     3.269 

3-301    3-333    3-365  3-398  3-431 

32 

2.1 

3.464    3.497     3.530    3.563     3.597 

3.631    3.664   3.698  3.733  3.767 

34 

2.2 

3.801     3.836     3.871     3.906    3.941 

3.976   4.012   4.047  4.083  4.119 

35 

2-3 

4.155     4.191     4.227    4.264    4.301 

4-337   4-374   4-412  4-449  4.486 

36 

2.4 

4.524    4.562     4.600    4.638    4.676 

4-714   4-753   4.792  4-831  4-870 

38 

2-5 

4.909    4.948     4.988     5.027     5.067 

5.107    5.147    5.187  5.228  5.269 

40 

2.6 

5.309     5-350     5.391     5.433     5.474 

5.515    5-557    5-599  5-641  5-683 

41 

2.7 

5.726     5.768     5.8II     5.853     5.896 

5.940   5.983    6.0266.0706.114 

43 

2.8 

6.158     6.2O2     6.246    6.290    6.335 

6.379    6.424   6.469  6.514  6.560 

44 

2.9 

6.605     6.651     6.697    6.743     6.789 

6.835    6.881    6.928  6.975  7.022 

46 

3.0 

7.069     7.II6     7.163     7.2II     7.258 

7-306   7.354   7.402  7.451   7.499 

48 

3-i 

7.548     7.596     7.645     7.694    7.744 

7-793    7.843    7-892  7.942  7.992 

49 

3-2 

8.042     8.093     8.143     8.194    8.245 

8.296   8.347   8.398  8.450  8.501 

51 

3-3 

8.553     8.605     8.657     8.709    8.762 

8.814   8.867    8.920  8.973  9.026 

52 

3-4 

9.079    9.133     9.186    9.240    9.294 

9.348    9.402   9.457  9.511  9.566 

54 

3-5 

9.621     9.676    9.731     9.787     9.842 

9.898     9-954     10.01    10.07    10.12 

56 

3-6 

10.18    10.24    10.29    IO-35    10.41 

10.46   10.52   10.58  10.64  10.69 

6 

3-7 

10.75    10.81    10.87    10.93    10.99 

11.04     II.  10     II.  l6    11.22    11.28 

6 

3-8 

11.34    11.40    11.46    11.52    11.58 

11.64    1^-70    11.76  11.82  11.88 

6 

3  9 

11.95    12.01    12.07    12.13    12.19 

12.25    12.32    12.38  12.44  12.50 

6 

4.0 

12.57    12.63    12.69    12.76    12.82 

12.88    12.95    13.01  13.07  13.14 

7 

4.1 

13.20    13.27    13.33    13-40    13.46 

13-53    I3.59    13.66  13.72  13.79 

7 

4.2 

13.85    13.92    13.99    14.05    14-12 

14-19    14-25    14-32  14-39  14-45 

7 

43 

14.52    14.59    14.66    14.73    14-79 

14.86    14.93    15.00  15.07  15.14 

7 

4.4 

15.21    15.27    15.34    15.41    15-48 

15-55    15-62    15.69  15.76  15.83 

7 

4-5 

15.90    15.98    16.05    16.12    16.19 

16.26    16.33    16.40  16.47  16.55 

7 

4.6 

16.62    16.69    16.76    16.84    16.91 

16.98    17.06    17.13  17.20  17.28 

7 

4-7 

17.35  17.42  17.50  17.57  17.65 

17.72    17-80    17.87  17.95  18.02 

8 

4.8  '18.10    18.17    18.25    18.32    18.40 

18.47    18.55    18.63  18.70  18.78 

8 

4.9 

18.86    18.93    19.01    19  09    19.17 

19.24   19.32    19.40  19.48  19.56 

8 

5-o 

19.63"  19.71  19.79  19.87  19.95 

20.03     20.11     20.19    20.27    20-35 

8 

5-1 

20.43  20.51   20.59   20.67  20.75 

20.83    20.91    20.99  21.07  21.  16 

8 

5-2 

21.24  21.32  21.40  21.48   21.57 

21.65  21.73  21.81  21.90  21.98 

8 

5-3 

>22.O6     22.15     22.23     22.31     22.4O 

22.48  22.56  22.65  22.73  22.82 

8 

5-4 

22.90     22.99    23.07     23.16     23.24 

23.33  23.41  23.50  23.59  23.67 

9 

d 

01234 

56789 

Diff. 

Explanation  in  Art.  196. 


MATHEMATICAL  TABLES 


571 


TABLE  51.     AREAS  OF  CIRCLES 


1  

d 

01234 

56789 

Diff. 

5.5 

23.76    23.84    23.93   24.02    24.11 

24.19   24.28    24.37  24.45  24.54 

9 

5-6 

24.63   24.72    24.81    24.89   24.98 

25.07    25.16  25-25  25.34  25.43 

9 

5-7 

25.52    25.61    25.70   25.79    25.88     25.97    26.06   26.15  26.24  26.33 

9 

5-8 

26.42    26.51    26.60   26.69    26.79    26.88    26.97    27.06  27.15  27.25 

9 

5-9 

27.34   27.43    27.53    27.62    27.71 

27.81    27.90   27.99  28.09  27.18 

9 

6.0 

28.27    28.37    28.46    28.56    28.65 

28.75    28.84   28.94  29.03  29.13 

9 

6.1 

29.22    29.32    29.42    29.51    29.61 

29.71    29.80   29.90  30.00  30.09 

10 

6.2 

30.19   30.29   30.39   30.48    30.58 

30.68    30.78    30.88  30.97  31.07 

10 

6.3 

31.17    31.27    31.37    31.47   31-57 

31.67    31.77    31.87  31.97  32.07 

10 

6-4 

32.17    32.27    32.37    32.47    32.57 

32.67    32.78    32.88  32.98  33.08 

10 

6.5 

33.18    33.29   33.39   33-49    33-59 

33-70   33.80   33-90  34-oo  34.11 

10 

6.6 

34.21    34.32    34.42   34.52   34.63 

34-73    34-84   34-94  35-Q5  35-15 

IO 

6-7 

35-26   35.36   35.47    35-57    35-68 

35-78    35.89   36.00  36.10  36.21 

IO 

6.8 

36.32    36.42   36.53   36.64   36.75 

36.85    36.96   37.07  37.18  37.28 

II 

6.9 

37-39   37-50   37.61    37.72   37.83 

37.94   38.05    38.16  38.26  38.37 

II 

7.0 

38.48    38.59   38.70   38.82    38.93 

39.04   39.15    39-26  39.37  39.48 

II 

7-1 

39-59   39-70   39.82    39.93   40.04 

40.15   40.26   40.38  40.49  40.60 

II 

7.2 

40.72   40.83   40.94   41.06   41.17 

41.28   41.40   41.51  41-62  41.74 

II 

7-3 

41.85    41.97   42-08   42.20  42.31 

42.43   42-54  42.66  42.78  42.89 

II 

7-4 

43.01    43.12   43.24  43.36   43.47 

43-59   43-71    43.83  43-94  44-o6 

12 

7-5 

44.18   44.30   44.41    44.53   44-65 

44.77   44.89   45.01  45.13  45-25 

12 

7.6 

45.36   45.48   45.60   45.72   45.84 

45.96   46.08   46.20  46.32  46.45 

12 

7-7 

46.57   46.69   46.81    46.93   47.05 

47.17   47-29   47-42  47-54  47-66 

12 

7-8 

47.78   47.91    48.03   48.15    48.27 

48.40  48.52    48.65  48.77  48-89 

12 

7-9 

49.02   49.14   49.27   49.39   49-51 

49.64   49.76   49-89  So.oi  50.14 

12 

8.0 

50.27    50.39    50.52    50.64    50.77 

50.90  51.02    51.15  51.28  51.40 

13 

8.1 

51.53    51.66   51.78    51.91    52.04 

52.17    52-30   52.42  52.55  52.68 

13 

8.2 

52.81    52.94    53.07    53.20   53.33 

53-46    53-59    53.72  53-85  53-98 

13 

8-3 

54.11    54.24    54-37    54.50    54.63 

54.76    54.89    55.02  55.15  55.29 

13 

8.4 

55-42    55-55    55-68    55.81    55-95 

56.08    56.21    56.35  56.48  56.61 

13 

8-5 

56.75    56.88    57.01    57.15    57-28 

57-41    57.55    57.68  57.82  57-95 

13 

8.6 

58.09    58.22    58.36    58.49    58.63 

58.77    58-90   59-04  59-17  59-31 

14 

8-7 

59.45    59.58    59.72    59.86    59.99 

60.13    60.27   60.41  60.55  60.68 

14 

8.8 

60.82   60.96   61.10   61.24   61.38 

61.51    61.65    61.79  61.93  62.07 

14 

8.9 

62.21    62.35    62.49  62.63   62.77 

62.91    63.05    63.19  63.33  63.48 

14 

9.0 

63.62   63.76   63.90   64.04   64.18 

64.33    64.47   64.61  64.75  64.90 

14 

9.1 

65.04   65.18   65.33   65.47    65.61 

65.76   65.90   66.04  66.19  66.33 

14 

9.2 

66.48  66.62   66.77   66.91    67.06 

67.20   67.35   67.49  67.64  67.78 

15 

9-3 

67.93   68.08    68.22   68.37   68.51 

68.66   68.  81    68.96  69.10  69.25 

15 

9-4 

69.40   69.55    69.69   69.84   69.99 

70.14    70.29   70.44  70.58  70.73 

15 

9-5 

70.88    71.03    71.18    71.33    71.48 

71.63    71.78    71.93  72.08  72.23 

15 

9-6 

72.38    72.53    72.68    72.84   72.99 

73.14    73.29   73.44  73.59  73-75 

15 

9-7 

73.90   74.05  -74-20   74.36    74.51 

74.66    74.82    74.97  75.12  75.28 

15 

9.8 

75-43    75.58    75.74   75.89    76.05 

76.20   76.36    76.51  76.67  76.82 

16 

9.9 

76.98    77-13    77-29    77-44    77-60 

77-76    77.91    78.07  78.23  78.38 

16 

d 

01234 

56789 

Diff. 

Explanation  in  Art.  196. 


572  MATHEMATICAL  TABLES 

TABLE  52.     TRIGONOMETRIC  FUNCTIONS 


Angle 

Arc 

Sin 

Tan 

Sec 

Cosec 

Cot 

Cos 

Coarc 

O 

0. 

0. 

O. 

!. 

00 

00 

I. 

1.5708 

90 

I 

0.0175 

0.0175 

0.0175 

I.OOOI 

57-299 

57.290 

0.9998 

•5533 

89 

2 

•0349 

•0349 

•0349 

1.  0006 

28.654 

28.636 

•9994 

•5359 

88 

3 

.0524 

•0523 

.0524 

1.0014 

19.107 

19.081 

.9986 

.5184 

87 

4 

.0698 

.0698 

.0699 

1.0024 

H.336 

14.301 

.9976 

.5010 

86 

5 

.0873 

.0872 

.0875 

1.0038 

11.474 

11.430 

.9962 

.4835 

85 

6 

0.1047 

0.1045 

O.I05I 

1.0055 

9.5668 

9.5144 

0-9945 

1.4661 

84 

7 

.1222 

.1219 

.1228 

1.0075 

8.2055 

8.1443 

.9925 

.4486 

83 

8 

.1396 

.1392 

.1405 

1.0098 

7.1853 

7.1154 

.9903 

•4312 

82 

9 

.1571 

.1564 

.1584 

1.0125 

6.3925 

6.3138 

.9877 

•4137 

81 

10 

•I745 

.1736 

.1763 

1.0154 

5.7588 

5.6713 

.9848 

.3963 

80 

ii 

0.1920 

0.1908 

0.1944 

1.0187 

5.2408 

5.1446 

0.9816 

L3788 

79 

12 

.2094 

.2079 

.2126 

1.0223 

4.8097 

4.7046 

.978i 

•3614 

78 

13 

.2269 

•2250 

.2309 

1.0263 

4-4454 

4.3315 

•9744 

•3439 

77 

14 

•2443 

.2419 

•2493 

1.0306 

4.I336 

4.0108 

.9703 

.3264 

76 

15 

.2618 

.2588 

.2679 

1.0353 

3-8637 

3.7321 

•9659 

.3090 

75 

16 

0.2793 

0.2756 

0.2867 

1.0403 

3.6280 

34874 

0.9613 

1.2915 

74 

i? 

.2967 

.2924 

.3057 

L0457 

3-4203 

3.2709 

•9563 

.2741 

73 

18 

•3142 

.3090 

.3249 

1-0515 

3-2361 

3.0777 

•9511 

.2566 

72 

19 

•33i6 

.3256 

•3443 

1.0576 

3.0716 

2.9042 

•9455 

.2392 

7i 

20 

•3491 

.3420 

.3640 

1.0642 

2.9238 

2-7475 

•9397 

.2217 

70 

21 

0.3665 

0.3584 

0-3839 

1.0712 

2.7904 

2.6051 

0.9336 

1.2043 

69 

22 

.3840 

.3746 

.4040 

1.0785 

2.6695 

2-4751 

.9272 

.1868 

68 

23 

.4014 

.3907 

•4245 

1.0864 

2-5593 

2-3559 

•9205 

.1694 

67 

24 

.4189 

.4067 

•4452 

1.0946 

2.4586 

2.2460 

.9135 

•1519 

66 

25 

•4363 

.4226 

.4663 

1.1034 

2.3662 

2.H45 

.9063 

•1345 

65 

26 

0-4538 

0.4384 

0.4877 

1.1126 

2.2812 

2.0503 

0.8988 

i  .  1  1  70 

64 

27 

.4712 

•4540 

.5095 

1.1223 

2.2027 

.9626 

.8910 

.0996 

63 

28 

.4887 

•4695 

•5317 

1.1326 

2.1301 

.8807 

.8829 

.0821 

62 

29 

.5061 

.4848 

•5543 

LI434 

2.0627 

8040 

.8746 

.0646 

61 

30 

•5236 

.5000 

•5774 

I-I547 

2.0000 

.7321 

.8660 

.0472 

60 

31 

0.5411 

0.5150 

0.6009 

1.1666 

.9416 

.6643 

0.8572 

1.0297 

59 

32 

.5585 

•5299 

.6249 

1.1792 

.8871 

.6003 

.8480 

1.0123 

58 

33 

.5760 

.5446 

.6494 

1.1924 

.8361 

•5399 

.8387 

0.9948 

57 

34 

•5934 

•5592 

•6745 

1.2062 

.7883 

.4826 

.8290 

•9774 

56 

35 

.6109 

.5736 

.7002 

1.2208 

•7435 

.4281 

.8192 

•9599 

55 

36 

0.6283 

0.5878 

0.7265 

1.2361 

.7013 

•3764 

0.8090 

0.9425 

54 

37 

.6458 

.6018 

•7536 

1.2521 

.6616 

.3270 

.7986 

.9250 

53 

38 

.6632 

-6I57 

•  7813 

1.2690 

.6243 

•2799 

.7880 

.9076 

52 

39 

.6807 

.6293 

.8098 

1.2868 

-5890 

•2349 

.7771 

.8901 

51 

40 

.6981 

.6428 

.8391 

L3054 

•5557 

.1918 

.7660 

.8727 

50 

41 

0.7156 

0.6561 

0.8693 

1-3250 

.5243 

•1504 

0.7547 

0.8552 

49 

42 

•7330 

.6691 

.9004 

I-3456 

•4945 

.1106 

•7431 

.8378 

48 

43 

.7505 

.6820 

•9325 

1-3673 

.4663 

.0724 

.7314 

.8203 

47 

44 

.7679 

.6947 

.9657 

1.3902 

.4396 

•0355 

•7193 

.8028 

46 

45 

.7854 

.7071 

i. 

1.4142 

.4142 

.7071 

.7854 

45 

Coarc 

Cos 

Cot 

Cosec 

Sec 

Tan 

Sin 

Arc 

Angle 

Explanation  in  Art.  196. 


MATHEMATICAL  TABLES 


573 


TXBLE   53.     LOGARITHMS   OF   TRIGONOMETRIC   FUNCTIONS 


Angle 

Log  Arc 

Log  Sin 

Log  Tan 

Log  Sec 

Log 
Cosec 

Log  Cot 

Log  Cos 

Log 
Coarc 

0 

—  CO 

—  oo 

—  00 

0. 

00 

00 

0. 

0.1961 

90 

I 

2.2419 

2.2419 

2.2419 

0.0001 

1.7581 

I.758I 

T.9999 

.1913 

89 

2 

.5429 

.5428 

•5431 

.0003 

•4572 

.4569 

•9997 

.1864 

88 

3 

.7190 

.7188 

.7194 

.0006 

.2812 

.2806 

•9994 

.1814 

87 

4 

.8439 

.8436 

.8446 

.001  1 

.1564 

•1554 

.9989 

.1764 

86 

5 

.9408 

•9403 

.9420 

.0017 

•0597 

.0580 

.9983 

.1713 

85 

6 

T.O2OO 

1.0192 

1.  0216 

0.0024 

0.9808 

0.9784 

1.9976 

0.1662 

84 

7 

.0870 

.0859 

.0891 

.0032 

.9141 

.9109 

.9968 

.1610 

83 

8 

•1450 

.1436 

.1478 

.0042 

.8564 

.8522 

.9958 

•1557 

82 

9 

.1961 

•1943 

.1997 

.0054 

.8057 

.8003 

.9946 

.1504 

81 

10 

.2419 

•2397 

.2463 

.0066 

.7603 

•7537 

•9934 

.1450 

80 

1  1 

1.2833 

1.2806 

I  2887 

0.0081 

0.7194 

0.7II3 

1.9919 

0.1395 

79 

12 

.3211 

•3*79 

.3275 

.0096 

.6821 

.6725 

.9904 

.1340 

78 

T3 

.3558 

•3521 

.3634 

.0113 

.6479 

.6366 

•9887 

.1284 

77 

14 

.3880 

.3837 

.3968 

.0131 

.6163 

.6032 

.9869 

.1227 

76 

15 

.4180 

•4130 

.4281 

.0151 

.5870 

.5719 

.9849 

.1169 

75 

16 

1.4460 

1.4403 

1.4575 

0.0172 

0-5597 

0.5425 

1.9828 

0.  1  1  1  1 

74 

17 

.4723 

.4659 

.4853 

.0194 

•5341 

.5M7 

.9806 

.1052 

73 

18 

.4972 

.4900 

.5118 

.0218 

.5100 

.4882 

.9782 

.0992 

72 

19 

.5206 

.5126 

•5370 

.0243 

.4874 

.4630 

•9757 

.0931 

7i 

20 

.5429 

•5341 

.5611 

.0270 

.4659 

.4^89 

•9730 

.0870 

70 

21 

1.5641 

1-5543 

1.5842 

0.0298 

0.4457 

0.4158 

1.9702 

0.0807 

69 

22 

.5843 

.5736 

.6064 

.0328 

.4264 

.3936 

.9672 

.0744 

68 

23 

.6036 

•5919 

.6279 

.0360 

.4081 

.3721 

.9640 

.0680 

67 

24 

.6221 

.6093 

.6486 

.0393 

.3907 

.35H 

.9607 

.0614 

66 

25 

.6398 

.6259 

.6687 

.0427 

•3741 

.3313 

•9573 

.0548 

65 

26 

1.6569 

1.6418 

1.6882 

0.0463 

0.3582 

0.3II8 

1-9537 

0.0481 

64 

27 

.6732 

.6570 

.7072 

.0501 

•3430 

.2928 

•9499 

.0412 

63 

28 

.6890 

.6716 

.7257 

.0541 

•3284 

•2743 

•9459 

•0343 

62 

29 

.7042 

.6856 

.7438 

.0582 

.3144 

.2562 

.9418 

.0272 

61 

30 

.7190 

.6990 

.7614 

.0625 

.3010 

.2386 

•9375 

.O2OO 

60 

31 

1-7332 

1.7118 

1.7788 

0.0669 

0.2882 

O.22I2 

I-933I 

O.OI27 

59 

32 

.7470 

•  7242 

.7958 

.0716 

.2758 

.2O42 

•9284 

0.0053 

58 

33 

.7604 

.7361 

.8125 

.0764 

•2639 

.1875 

.9236 

1.9978 

57 

34 

•7734 

.7476 

.8290 

.0814 

•2524 

.I7IO 

.9186 

.9901 

56 

35 

.7859 

.7586 

.8452 

.0866 

.2414 

.1548 

.9134 

.9822 

55 

36 

1.7982 

1.7692 

Y.86I3 

0.0920 

0.2308 

0.1387 

1.9080 

1-9743 

54 

37 

.8101 

•7795 

.8771 

.0977 

.2205 

.1229 

.9023 

.9662 

53 

38 

.8217 

.7893 

.8928 

.1035 

.2107 

.1072 

.8965 

•9579 

52 

39 

.8329 

.7989 

.9084 

.1095 

.2OI  I 

.0916 

.8905 

•9494 

5i 

40 

.8439 

.8081 

.9238 

.1157 

.1919 

.0762 

.8843 

.9408 

50 

4i 

1.8547 

1.8169 

1.9392 

0.1222 

0.1831 

O.O6O8 

1.8778 

1.9321 

49 

42 

.8651 

.8255 

•9544 

.1289 

•1745 

.0456 

.8711 

.9231 

48 

43 

.8754 

.8338 

.9697 

•1359 

.1662 

.0303 

.8641 

.9140 

47 

44 

•8853 

.8418 

.9848 

.1431 

.1582 

.0152 

.8569 

.9046 

46 

45 

•8951 

.8495 

o. 

.1505 

.1505 

O. 

.8495 

.8951 

45 

Log 
Coarc 

Log  Cos 

Log  Cot 

Log 
Cosec 

Log  Sec 

Log  Tan 

Log  Sin 

Log  Arc 

Angle 

Explanation  in  Art.  196. 


574 


MATHEMATICAL  TABLES 


TABLE  54.     LOGARITHMS  OF  NUMBERS 


n 

01234 

56789 

Diff. 

10 

oooo  0043  0086  0128  0170 

0212   0253   0294   0334   0374 

42 

II 

0414  0453  0492  0531  0569 

0607   0645   0682   0719   0755 

38 

12 

0792  0828  0864  0899  0934 

0969  1004  1038  1072  1106 

35 

13 

"39  II7.3  1206  1239  I27* 

1303  1335  1367  1399  1430 

32 

14 

1461  1492  1523  1553  1584 

1614  1644  1673  1703  1732 

30 

15 

1761  1790  1818  1847  1875 

1903  1931  1959  1987  2014 

28 

16 

2041   2068   2095   2122   2148 

2175   2201   2227   2253   2279 

27 

17 

2304  2330  2355  2380  2405 

2430   2455   2480   2504   2529 

25 

18 

2553  2577  2601  2625  2648 

2672   2695   27l8   2742   2765 

24 

19 

2788  2810  2833  2856  2878 

2900   2923   2945   2967   2989 

22 

20 

3010  3032  3054  3075  3096 

3Il8   3139   3160   3l8l   3201 

21 

21 

3222  3243  3263  3284  3304 

3324  3345  3365  3385  3404 

20 

22 

3424  3444  3464  3483  3502 

3522  3541  3560  3579  3598 

19 

23 

3617  3636  3655  3674  3692 

3711  3729  3747  3766  3784 

18 

24 

3802  3820  3838  3856  3874 

3892  3909  3927  3945  3962 

18 

25 

3979  3997  4014  4031  4048 

4065  4082  4099  4116  4133 

17 

26 

4150  4166  4183  4200  4216 

4232  4249  4265  4281  4298 

17 

27 

4314  4330  4346  4362  4378 

4393  4409  4425  4440  4456 

16 

28 

4472  4487  4502  4518  4533 

4548  4564  4579  4594  4609 

15 

29 

4624  4639  4654  4669  4683 

4698  4713  4728  4742  4757 

15 

30 

4771  4786  4800  4814  4829 

4843  4857  4871  4886  4900 

14 

31 

4914  4928  4942  4955  4969 

4983  4997  Son  5024  5038 

14 

32 

5051  5065  5079  5092  5105 

5119  5132  5M5  5159  5172 

13 

33 

5185  5198  5211  5224  5237 

5250  5263  5276  5289  5302 

13 

34 

5315  5328  5340  5353  5366 

5378  5391  5403  54i6  5428 

13 

35 

5441  5453  5465  5478  5490 

5502  5514  5527  5539  5551 

12 

36 

5563  5575  5587  5599  56n 

5623  5635  5647  5658  5670 

12 

37 

5682  5694  5705  5717  5729 

5740  5752  5763  5775  5786 

12 

38 

5798  5809  5821  5832  5843 

5855  5866  5877  5888  5899 

II 

39 

5911  5922  5933  5944  5955 

5966  5977  5988  5999  6010 

II 

40 

6021  6031  6042  6053  6064 

6075  6085  6096  6107  6117 

II 

4i 

6128  6138  6149  6160  6170 

6180  6191  6201  6212  6222 

II 

42 

6232  6243  6253  6263  6274 

6284  6291  6304  6314  6325 

10 

43 

6335  6345  6355  6365  6375 

6385  6395  6405  6415  6425 

IO 

44 

6435  6444  6454  6464  6474 

6484  6493  6503  6513  6522 

IO 

45 

6532  6542  6551  6561  6571 

6580  6590  6599  66°9  6618 

10 

46 

6628  6637  6646  6656  6665 

6675  6684  6693  6702  6712 

9 

47 

6721  6730  6739  6749  6758 

6767  6776  6785  6794  6803 

9 

48 

6812  6821  6830  6839  6848 

6857  6866  6875  6884  6893 

9 

49 

6902  6911  6920  6928  6937 

6946  6955  6964  6972  6981 

9 

50 

6990  6998  7007  7016  7024 

7033  7042  7050  7059  7067 

9 

51 

7076  7084  7093  7101  7110 

7118  7126  7135  7143  7152 

8 

52 

7160  7168  7177  7185  7193 

7202  7210  7218  7226  7235 

8 

53 

7243  7251  7259  7267  7275 

7284  7292  7300  7308  7316 

8 

54 

7324  7332  7340  7348  7356 

7364  7372  738o  7388  7396 

8 

n 

01234 

5     6     7    8    9 

Diff. 

Explanation  in  Art.  8. 


MATHEMATICAL  TABLES 


575 


TABLE  54.     LOGARITHMS  OF  NUMBERS 


n 

01234 

56789 

Diff. 

55 

7404  7412  7419  7427  7435 

7443  7451  7459  7466  7474 

8 

56 

7482  7490  7497  7505  7513 

7520  7528  7536  7543  7551 

57 

7559  7566  7574  7582  7589 

7597  7604  7612  7619  7627 

58 

7634  7642  7649  7657  7664 

7672  7679  7686  7694  7701 

59 

7709  7716  7723  7731  7738 

7745  7752  776o  7767  7774 

60 

7782  7789  7796  7803  7810 

7818  7825  7832  7839  7846 

7 

61 

7853  7860  7868  7875  7882 

7889  7896  7903  7910  7917 

62 

7924  7931  7938  7945  7952 

7959  7966  7973  7980  7987 

63 

7993  8000  8007  8014  8021 

8028  8035  8041  8048  8055 

64 

8062  8069  8075  8082  8089 

8096  8102  8109  8116  8122 

65 

8129  8136  8142  8149  8156 

8162  8169  8176  8182  8189 

7 

66 

8195  8202  8209  8215  8222 

8228  8235  8241  8248  8254 

67 

8261  8267  8274  8280  8287 

8293  8299  8306  8312  8319 

68 

8325  8331  8338  8344  8351 

8357  8363  8370  8376  8382 

69 

8388  8395  8401  8407  8414 

8420  8426  8432  8439  8445 

70 

8451  8457  8463  8470  8476 

8482  8488  8494  8500  8506 

6 

71 

8513  8519  8525  8531  8537 

8543  8549  8555  8561  8567 

72 

8573  8579  8585  8591  8597 

8603  8609  8615  8621  8627 

73 

8633  8639  8645  8651  8657 

8663  8669  8675  8681  8686 

74 

8692  8698  8704  8710  8716 

8722  8727  8733  8739  8745 

75 

8751  8756  8762  8768  8774 

8779  8785  8791  8797  8802 

6 

76 

8808  8814  8820  8825  8831 

8837  8842  8848  8854  8859 

77 

8865  8871  8876  8882  8887 

8893  8899  8904  8910  8915 

78 

8921  8927  8032  8938  8943 

8949  8954  8960  8965  8971 

79 

8976  8982  8987  8993  8998 

9004  9009  9015  9020  9025 

80 

9031  9036  9042  9047  9053 

9058  9063  9069  9074  9079 

5 

81 

9085  9090  9096  9101  9106 

9112  9117  9122  9128  9133 

82 

9138  9143  9149  9J54  9X59 

9165  9170  9175  9180  9186 

83 

9191  9196  9201  9206  9212 

9217  9222  9227  9232  9238 

84 

9243  9248  9253  9258  9263 

9269  9274  9279  9284  9289 

85 

9294  9299  9304  9309  9315 

9320  9325  9330  9335  9340 

5 

86 

9345  9350  9355  9360  9365 

9370  9375  9380  9385  9390 

87 

9395  94°°  9405  9410  94*5 

9420  9425  9430  9435  9440 

88 

9445  9450  9455  9460  9465 

9469  9474  9479  9484  9489 

89 

9494  9499  9504  9509  9513 

9518  9523  9528  9533  9538 

90 

9542  9547  9552  9557  9562 

9566  9571  9576  9581  9586 

5 

91 

9590  9595  96°o  9605  9609 

9614  9619  9624  9628  9633 

92 

9638  9643  9647  9652  9657 

9661  9666  9671  9675  9680 

93 

9685  9689  9694  9699  9703 

9708  9713  9717  9722  9727 

94 

9731  9736  9741  9745  9750 

9754  9759  9763  9768  9773 

95 

9777  9782  9786  9791  9795 

9800  9805  9809  9814  9818 

4 

96 

9823  9827  9832  9836  9841 

9845  9850  9854  9859  9863 

97 

9868  9872  9877  9881  9886 

9890  9894  9899  9903  9908 

98 

9912  9917  9921  9926  9930 

9934  9939  9943  9948  9952 

99 

9956  9961  9965  9969  9974 

9978  9983  9987  9991  9996 

n 

01234 

56789 

Diff. 

Explanation  in  Art.  8. 


576 


MATHEMATICAL  TABLES 


TABLE  55.     CONSTANTS  AND  THEIR  LOGARITHMS 


Name. 
(Radius  of  circle  or  sphere  =  i.) 

Symbol. 

Number. 

Logarithm. 

Area  of  circle 

it 

3.141  592654 

0.497149873 

Circumference  of  circle 

27T 

6.283  185  307 

0.798  179868 

Surface  of  sphere 

op 

12.566370614 

1.099209  864 

i* 

0.523598776 

1.718998622 

Quadrant  of  circle 

%n 

0.785398  163 

1.895089881 

Area  of  semicircle 

t* 

1.570796327 

0.196  119877 

Volume  of  sphere 

1* 

4.187790205 

O.622  088  609 

7TS 

9.869604401 

0.994299745 

n* 

1.772453851 

0.248  574936 

Degrees  in  a  radian 

I80/7T 

57-295  779513 

1.758  122  632 

Minutes  in  a  radian 

I0800/7T 

3437-74677I 

3.536273883 

Seconds  in  a  radian 

648ooo/7T 

206  264.806 

5.314425  133 

I/* 

0.318  309886 

1.502  850  127 

I/** 

0.564  189584 

1.751425064 

It* 

o.  101  321  184 

1.005  700255 

Circumference/36o 

arc    ° 

0.017453293 

2.241877368 

sin    ° 

0.017  452  406 

2.241  855  318 

Ci  rcumference/2  1  600 

arc    ' 

o.ooo  290  888 

4.463  726  117 

sin    ' 

o.ooo  290  888 

4.463  726  III 

Circumference/I296ooo 

arc    " 

o  .  ooo  004  848 

6.685  574867 

sin    " 

o  .  ooo  004  848 

6.685  574867 

Base  Naperian  system  of  logs 

e 

2.718  281  828 

0.434294482 

Modulus  common  system  of  logs 

M 

0.434294482 

1.637  7843H 

Naperian  log  of  10 

i/M 

2.302  585  093 

0.362  215  689 

hr 

0.4769363 

1.6784604 

Probable  error  constant 

hr  |/2~ 

0.6744897 

1.8289754 

Feet  in  one  meter 

m/ft. 

3.2808333 

0.5159841 

Miles  in  one  kilometer 

km/mi. 

0.621  3699 

1-7933502 

INDEX 


577 


INDEX 


Absolute  velocity,  69,  74,  402,  426 

Acceleration,  4,  13,  20,  548 

Acre-foot,  350 

Adams,  A.  L.,  256 

Adjutage,  178,  186 

Advantageous  nozzle,  431 

section,  276,  279 
velocity,    402,    417, 
45i,  5i6 

Air  chamber,  235,  394,  503 

Air-lift  pump,  523 

Air  valve,  218 

Anchor  ice,  6 

Answers  to  problems,  540 

Approach,  angle  of,  426,  450 

velocity  of,  56,  124,  146 
168,    153 

Aqueducts,  i,  204,  265 

Archimedes,  i,  26,  495 

Areas  of  circles,  16,  560 

Atmospheric  pressure,  2,  9,  19,  25, 
44,  498,   547 

Backpitch  wheel,  433 
Backwater,  326,  334,  335 

function,  336,  566 
Ball  nozzle,  193 
Barker's  mill,  437 
Barometer,  9,  19,  498 
Bazin,   H.,   2,   105,   159,   169,    200, 

274,  565 

Bazin's  formula,  290,  303 
Bernouilli,  D.,  2,  76,  186,  196 
Bidone,  G.,  2,  129,  343,  378 
Blow-offs,  218 
Bodmer,  G.  R.,  445 
Boiling  point,  10,  498 
Bore,  344,  345 
Bossut,  C.,  2,  112,  113 
Buoyancy,  center  of,  30 
Bowie,  A.  J.,  122,  423 


Boy  den,  U.,  84,  464 
Boyden  diffuser,  463 

hook  gage,  84 

turbine,  370,  447 
Bramah,  J.,  22 
Branched  pipes,  249 
Bresse,  M.,  336,  339 
Breast  wheels,  418,  434 
Brick  conduits,  282,  293 

sewers,  285 
Brooks,  265,  306 
Browne,  R.  E.,  423 
Buckets,  417 

Bucket  pumps,  i,  495,  523 
Buff,  H.,  183 

Canals,  265,  286,  339,  341,  351 
Canal  boat,  479 

lock,  128,  137 
Carpenter,  R.  C.,  390 
Cascade  wheel,  423 
Castel,  183,  213 
Castelli,  B.,  2 

Cast-iron  pipes,  204,  261,  559,  560 
Center  of  buoyancy,  30,  488 
of  gravity,  32 
of  pressure,  35,  38 
Centrifugal  force,  71 

pump,  514 
Chain  pximp,  i,  495 
Channels,   265,   280,  302,  386,  565 
Chezy,  2 

Chezy's  formula,  261,  269,  301 
Cippoletti,  165,  1 66 
Circles,  areas  of,  542,  570 

properties  of,  273,  558 
Circular    conduit,     266,     270-275, 

301,  561-564 
orifices,  52,  116,  139,  551 
Classification  of  pumps,  496,  521 
of  surfaces,  282,  290 


578 


INDEX 


Classification  of  turbines,  439 

Coal  used  by  steamers,  480 

Coefficient  of  contraction,  in,  179 
in  nozzles,  190 
in  orifices,  112,  130 
in  tubes,  176,  179 

Coefficient  of  discharge,  115 
in  nozzles,  191 
in  orifices,  115-121,  127, 

130,132 
in  pipes,  195 
in  tubes,  179,  195,  206 
in  turbines,  448 
in  weirs,  145-169 

Coefficient  of  roughness,  282,  290 

Coefficient  of  velocity,  112,  179 
in  orifices,  113 
in  nozzles,  191,  240 
in  tubes,  179,  195 

Cole,  E.  S.,  93,  200 

Compound  pipes,  237 
tubes,  1 86 

Compressibility  of  water,  n,  20 

Computations,  14,  17 

Conduit  pipes,  255,  289 

Conduits,  265-304,  561-564 

Conical  tubes,  183,  558 
wheel,  434 

Conservation  of  energy,  76,  188 

Constants,  tables  of,  543-576 

Consumption  of  water,  352 

Contracted  weirs,  141,  148,  554 

Contraction  of  a  jet,  2,  no 

coefficient  of,  in 
gradual,  177 
sudden,  176 
suppression  of,  128 

Cooper,  T.,  386 

Cotton  hose,  257 

Crest  of  a  weir,  85,  141 

rounded  and  wide,  156 

Critical -velocity,  262 

Ctesibius,  i,  495 

Cubic  foot,  4,  8,  546 

Current  indicators,  312 

meters,  99,  312,  320 

Curvature  factors,  213 


Curve  of  backwater,  161,  335 
Curved  surfaces,  34 
Curves  in  pipes,  212,  386 
in  rivers,  309,  388 

Darcy,  H.,  2,  105,  260,  274 
Dams,  32,  41,  45,  157,  325,  557 
Danaide  434 

Data,  fundamental,  1-21,  543 
D'Aubisson,  J.  F.,  183,  260,  328 
Depth  of  flotation,  27 
Design  of  turbines,  453 

of  power  plants,  356 
of  water  wheels,  435 
Diameters  of  pipes,  223 

water  mains,  248,  354 
Differential  pressure  gages,  90 
Diffuser,  463 
Discharge,  51,  82,  97,  in 

coefficient  of,  115 

conduits,  265-304 

curves,  322 

nozzles,  191,  202 

orifices,  109-140 

tubes,  107-203 

pipes,  204-264 

rivers,  305-346 

turbines,  446 

weirs,  141-169 
Discharging  capacity,  228 
Disk  valve,  216 
Displacement  pumps,  521 
Distilled  water,  8,  19 
Ditches,  265,  286 
Diverging  tubes,  186 
Diversions,  250,  530,  534 
Downing,  S.,  135 
Downward-flow  wheels,  428 

turbines,  440,  458 
Draft  tube,  428,  472 
Drop-down  curve,  334 

function,  567 
Dropping  head,  135 
Du  Bois,  A.  J.,  465 
Dubuat,  N.,  2,  295 
Duplex  pump,  504 
Duty  of  pumps,  511 


INDEX 


579 


Duty  of  water,  351 
Dynamic  pressure,  67,  375—411 
Dynamo,  372,  470 
Dynamometer,  363 

Effective  head,  56,  124,  240,  360 
power,  356,  362 

Efficiency,  66,  356,  412 

of  jet,  133,  1 80,  400 
of  jet  propeller,  482 
of  motors,  355,  365 
of  moving  vanes,  400 
of  paddle  wheels,  484 
of  pumps,  497-528 
of  reaction  wheel,  411 
of  screw  propeller,  486 
of    turbines,    356,    362, 

441-464 

of  water    wheels,    356, 
407,    412,    414,   427- 

435.  484 

Egg-shaped  sewers,  284 
Elasticity  of  water,  n,  20 
Electric  analogies,  252,  534 

generators,  372,  469  . 
Elevations  by  barometer,  10 
Ellis,  T.  G.,  120 
Emptying  a  canal  lock,  137 

a  vessel,  59,  72 
Energy,  4,  5.  76»  J97 

loss  of,  133 

in  channels,  269,  300 

tubes,  170,  197 

of  a  jet,  65 
English  measures,  3 
Enlargement  of  section,  173,  296 
Entrance  angle,  426,  450 
Errors  in  computations,  15,  108 

in  measurements,  105,  132, 

142,  358 

Eureka  turbine,  367,  445 
Evaporation,  347 
Ewbank,  T.,  506 
Ewart,  P.,  380 
Exit  angle,  426,  450 
Expansion  of  section,  173 
Eytelwein,  J.  A.,  2,  112,  186,  195 


Faesch  and  Picard,  446 

Falling  bodies,  13,  21,  46,  538 

Fanning,  J,  T.,  121 

Filaments,  210,  267 

Filter  bed,  262,  539 

Fire  hose,  213,  257,  263,  530 

service,  248,  354 
Flad,  H.,  91 
Fletcher,  R.,  218,  390 
Flinn  and  Dyer,  166 
Floats,  310,  319 
Flotation,  depth  of,  27 

stability  of,  29,  488 
Flow,  dynamic  pressure  of,  375-410 
from  orifices,  48,  109-140 
over  dams,  157 
of  electricity,  252,  534 
in  canals  and  conduits,  265— 

3°4,  340 

in  rivers,  305-345 
through  pipes,  75,  104,  204- 

265,  289 

through  tubes,  170-203 
through  turbines,  446 
under  pressure,  53,  79 
Flynn,  P.,  284 
Foot  valve,  501,  505 
Foote,  A.  D.,  124 
Force  pump,  i,  496,  502,  506 
Forebay,  298,  340 
Foss,  F.  E.,  293 

Fourneyron  turbine,  3,  440,  463 
Francis,  J.  B.,  2,  149,  152,  154,  157^ 

186,  310,  368,  449 
Francis  turbine,  440 

float  formula,  311 
weir  formula,  152 
Freeman,  J.  R.,  103,  200,  213 
Free  surface,  6,  24 
Frictional  resistances,  46 

in  channels,  269,  294 
in  pipes,    195,    208,  477 
in  pumps,  501,  507 
in  turbines,  414,  438 
in  water  wheels,  414,  435 
of  ships,  475,  478 
Friction  brake,  363 


580 


INDEX 


Friction  factors,  210,  257,  559 

heads,  209,  226,  297,  560 
Frizell,  J.  P.,  524 
Fteley,  A.,  148,  149,  154,  155,  157 

Gages,  4,  84-93 

Gaging  flow,  83,  131,  142,  318 

of    rivers,     313,    316,    319, 

322«  35° 
Galileo,  2,  3 

Gallon,  4,  8,  545 

Ganguillet,  E.,  281 

Gasoline  differential  gage,  93 

Gate  of  a  turbine,  440,  442,  463 

Gates,  pressure  on,  40 

Gate  valve,  217 

Girard,  P.,  464 

Glacier,  flow  of,  294 

Governor,  471 

Gradient,  hydraulic,  233,  238 

Graphic  methods,  105 

Grassi,  G.,  n 

Gravity,  acceleration  of,  13,  20 

center  of,  32 
Ground  water,  350 
Guides,  443,  450,  456 

Head,  24,  87,  143,  206,  360 

and  pressure,    24,   43,   410, 

543-  549 

losses  of,  133,  1 80,  194,  294 
measurement  of,  82,  85,  131, 
360 

Heat  units,  511,  533 

Hering,  C.,  282 

Height  of  jets,  49,  180,  192 
.  Herschel;  C.,  94,  154,  255,  289 

Historical  notes,  i,  22,  204,  265 

Holyoke  tests,  369,  447 

Hook  gage,  6,  84,  131,  143,  307 

Horizontal  impulse  wheels,  424 

range  of  a  jet,  63,  193 

Horse-power,  5,  18,  355,  371 

Horseshoe  conduits,  293 

Hose,  213,  257,  263,  529 

House-service  pipes,  243 

Humphreys  and  Abbot,  308,  320 

Hunt  turbine,  444 


Hurdy-gurdy  wheel,  423 

Hydraulic  engine,  520 

gradient,  233,  238 
jump,  343 
mean  depth,  266 
motors,    355-373,    407- 

472.  523 
radius,  265 
ram,  517 

Hydraulics,  defined,  i,  7 

theoretical,  46-80 

Hydromechanics,  i,  392,  474 

Hydrometric  balance,  312 

pendulum,  312 

Hydrostatic  head,  24,  76,  87 

Hydrostatics,  i,  22-45 

Ice,  6,  18,  538 
Immersed  bodies,  384,  477 
Impact,  174,  210,  378 
Impeller  pump,  521 
Impulse,  67,  375 

turbines,  441,  461 

wheels,  423-432 
Inclined  tubes,  196 
Inertia,  movements  of,  39,  489 
Injector  pump,  78,  523 
Instruments,  81-103 
Inward-flow  turbines,  440,  459 
Inward-projecting  tubes,  184 
Irrigation,  hydraulics,  351 

Jet  propeller,  481 
Jet  pump,  523 
Jets,  63-69,  556 

contraction  of,  2,  no,  in 

energy  of,  65 

from  nozzles,  192,  431 

height  of,  113,  192 

impulse  of,  67,  397 

on  vanes,  310,  395 

path  of,  63 

range  of,  63,  64 
Jonval  turbine,  440 
Joukowsky,  N.,  393 
Jump,  334,  342 


INDEX 


581 


Keely  motor,  23 

Kilowatt,  372 

Kinetic  energy,  5,  76 

Knot,  474 

Kuichling,  E.,  217 

Kutter,  W.  R.,  2,  281,  564 

Kutter's formula,  261,  281,303,306 

Lampe's  formula,  261,  263 

Lead  pipes,  204 

Leakage,  359,  456,   501,  512 

Least  squares,  method  of,  107 

LefFel  turbine,  444,  445 

Lesbros,  J.  A.,  114,  129,  149 

Lift  pump,  499 

Lighthouses,  385 

Linen  hose,  257 

Log,  311,  474 

Logarithms,  15,  573-576 

Long  pipes,  226 

Loss  of  head,   133,   180,   194,   207, 

216,  294,  560 
in  contraction,  176 
in  curvature,  212 
in  entrance,  185,  207, 

226,  295 
in    expansion,     173, 

295.  5°7 
in  friction,  195,  208, 

247,  297,  507 
measurement   of,  82, 

132,  197,  230 

Loss  of  weight  in  water,  26 
Lowell  tests,  368 

Mariotte,  2,  500 

Mars,  water  on,  14 

Marx,  Wing,  and  Hoskins,  255,  256, 

289 

Masonry  dams,  32,  45 
Mathematical  constants,  568-576 
Mean  velocity,  52,  97,  199,  218,  267 
Measuring  instruments,  81-103 
Mercury,  9,  55,  135 
Mercury  gage,  88,  90 
Merriman,  M.,  392 
Metacenter,  30,  488 


Meters,  current,  99 

water,  93,  131,  319,  358 

Venturi,  94 
Metric  system,  5,  18,  43,  78,  137, 

167,   202,  262,  301 
Michelotti,  F.  D.,  112,  113,  139 
Mill  power,  371 
Miner's  inch,  122 
Module,  123 

Modulus  of  elasticity,  n,  20 
Moments  of  inertia,  39,  489 
Morosi,  J.,  379 
Motors,  357-370 
Mouthpiece,  186 
Moving  vanes,  399 

Naval  hydromechanics,  474-494 
Navigation  canals,  341,  479 
Negative  pressure,  77,  180 
Newton,  I.,  2,  13,  112 
Niagara  power  plants,  372,  465 

turbines,  440,  446 
Noble,  T.  A.,  256,  289 
Non-uniform  flow,  329 
Normal  pressure,  31 
Nozzles,  189,  202,  240,  390,  430,  530 

jets  from,  103,  191,  242 
Numerical  computations,  14 

Oar,  action  of,  483 
Observations,  discussion  of,   105 
Obstructions  in  channels,  295 

in  pipes,  217 

Ocean  waves,  373,  386,  435 
Oil,  55,  91 

Oil  differential  gage,  91 
Orifices,  .48-6 2,  109-140,  551-553 
Oscillations,  488,  538 
Outward-flow  turbine,  441,  443 
Overshot  wheels,  415,  433 

Paddle  wheels,  483 
Paraboloid,  73 
Pascal,  2,  9,  22,  495 
Path  of  a  jet,  63 
Patent  log,  475 
Peirce,  C.  S.,  14,  20 


582 


INDEX 


Pelton  water  wheel,  424 
Penstock,  356,  359,  466 
Physical  properties  of  water,  5 
Piers,  327 
Piezometer,  84,  198,  209,  229 

gagings,  230 
Pipes,  75,   104,   196,   204-264,  390, 

525>  559-560 
curves  in,  212,  386 
friction  factors  for,  211,  232 
friction  heads  for,  560 
smooth,  75 
Pitometer,  244 

Pitot's  tube,  96,  102,  200,  312 
Plates,  moving,  384,  477 
Plunger  pumps,  504 
Plympton,  G.  W.,  10 
Pneumatic  turbine,  464 
Poiseuille's  law,  261 
Poncelet,  J.  V.,  2,  114 
Poncelet  wheel,  421 
Potential  energy,  5 
Power,  5,  18,  355,  371 

dynamometer,  363 
Press,  hydraulic,  22 
Pressure,  center  of,  35,  38 

dynamic,  375-411 

energy  of,  170 

flow  under,  53,  79 

gages,  4,  86,  93 

horizontal,  33 

measurement  of,  84—93, 
230,  512 

negative,  77 

normal,  31 

of  waves,  385,  493 

on  dams,  32,  41 

on  pipes,  205,  232 

on  planes,  33,  35 

regulator,  245 

transmission  of,  22,  375 

unit  of,  2 
Pressure  gage,  4,  84-93 

head,   24,  43,  84,  90,  232 
Price  current  meter,  100 
Probable  errors,  132 
Prony,  G.  F.,  2,  363 


Propeller,  481,  485 
Propulsion,  work  in,  479 
Pumps,  i,  10,  495-533 
Pumping  through  hose,  529 
Pumping  through  pipes,  525 
Pumping  engines,  510 
Puppet  valve,  506 

Rafter,  G.  W.,  159,  160,  169 

Ram,  hydraulic,  518 
in  pipes,  390 

Range  of  a  jet,  63,  193 

Radius,  hydraulic,  265 

Rainfall,  347 

Rankine,  W.  J.  M.,  2,  3,  481 

Rating  a  meter,  94,  101 

Reaction,  67,  377 

experiments  on,  378 
turbines,  442,  453 
wheel,  410,  437,  4^2 

Reciprocating  pumps,  521 

Rectangular  conduits,  275,  562 

orifices,  52,   121,  126, 

i39»  553 
Reducer,  237 
Regulatgr,  245 
Relative  capacities  of  pipes,  228 

velocity,  69,  395 
Reservoirs,  83,  205,  353 
Resistance  of  plates,  477 

of  ships,  476 
Reversibility,  523 
Revolving  tubes,  409 

vanes,  402 

vessel,  71 
Reynolds,  O.,  262 
Ring  nozzle,  189,  192 
Rivers,  3°5-345>  388 
River  water,  8,19 
Riveted  pipes,  204,  254,  289 
Rochester  water  pipe,  239 
Rod  float,  310 
Rolling  of  a  ship,  31 
Roman  aqueducts,  i,  204,  265 

pipes,  i,  204,  239 
Rossetti,  G.,  8,  19 
Rotary  pumps,  521 


INDEX 


583 


Rounded  crests,  156 

orifices,  130 
Rudder,  action  of,  490 
Ruhlmann,  M.,  379 
Runoff,  348 

Salt  water,  7,  8,  19,  474 
Sand,  weight  in  water,  27 

filter  bed,  262 
Screens,  295,  297 
Screw  propeller,    485 

turbine,  464 
Seepage,  351 
Sewage,  9,  524 
Sewers,  9,  283,  303,  563-564 
Ships,  31,  385,  474-491 
Short  pipes,  195,  225 

tube,  178 
Siamese  joint,  530 
Siphon,  235 
Skin  of  water,  6,  84 
Slagg,  C.,  445 
Slip,  484,  486 
Slope,  204,  266,  306 
Smith,  H.,  Jr.,  2,  8,  118,  120,  127, 

149,  151,   184,  270,  272,  275 
Smooth  nozzle,  189,  191 

pipes,  104 

Sound,  velocity  of,  12,  19 
Specific  gravity,  27,  44 
Speed  of  wheels,  408,  417,  423 

of  ships,  475 

of  turbines,  441,  447 
Sphere,  29 
Square  vertical  orifices,    118,    139, 

552 

Squares,  table  of,  568 
Stability  of  dams,  42 

of  flotation,  29,  488 
Standard  orifice,  109 

tube,  178 
Standpipe,  206 
Statical  moment,  39 
Steamer,  coal  used  by,  480 
Stearns,  F.  P.,  148,  149,  157, 
Steady  flow,  75,  197,  329 
Steel  pipes,  204,  254,  289 


Stevenson,  T.,  318 
Storage  of  water,  353 
Strength  of  pipes,  35,  44 
Submerged  bodies,  26 
orifices,  127 
surfaces,  477 
turbines,  442 
weirs,  154,  554 
Sub-surface  float,  310 

velocities,  308,  317 
Suction,  10,  495,  498 
Suction  pump,  499 
Sudbury  conduit,  283,  293,  300 
Suppressed  weirs,  141,151,.! 68,  555 
Suppression  of  contraction,  128 
Surface  curve,  161,  322,  335,  339 

velocity,  308,  317 

Surfaces,  center  of  pressure,  35^38 
jets  upon,  381,  395 
motion  of,  395 
pressure  on,  31,  35 

Tables,  viii,  16,  542-576 
Tank,  82,  126,  358 
Temperature,  6,  132,  498 
Test  of  motors,  362 

of  pumping  engines,  512 

of  turbines,  366,  470 
Thearle,  S.  J.  P.,  475 
Theoretical  hydraulics,  46-80 
Theoretic  discharge,  51 

velocity,  48,  50 
Thomson,  J.,  164,  389 
Throttle  valve,  171 
Thurston,  R.  H.,  366 
Tidal  bore,  345 
Tidal  waves,  373,  492 
Tide  gate,  49 
Tides,  325,  373,  435,  492 
Time,  4,  18 

Torricelli,  E.,  2,  48,  495 
Transmission  of  pressures,  22 
Transporting  capacity,  28$,  323 
Trapezoidal  conduits,  278 

weirs,  165 

Trautwinc,  J.  C.,  Jr.,  282 
Triangular  orifices,  no 


584 


INDEX 


Triangular  weirs,  163 
Trigonometric  functions,  572 
Triple  nozzle,  424 
Troughs,  265,  275 
Tubes,  170-203,  409 
Tunnel,  Niagara,  465 
Turbines,  356,  366,  437-472 
Twin  screws,  487 

turbines,  445 

Undershot  wheels,  420,  434 

Uniform  flow,  75,  197,  267 

Units  of  measure,  3,  18,  544 

Unsteady  flow,  321 

Unwin,  W.  C.,  132,  380 

U.  S.  Department  Agriculture,  322 
Geological  Survey,  322 
Weather  Bureau,  348 

Vacuum,  2,  9,  77,  188,  495,  498 
in  compound  tube,  188 
in  pumps,  498,  523 
in  standard  tube,  182 
in  turbines,  459 
Valves,  43,  216,  499,  505 
Vanes,  68,  395,  456 

in  motion,  399 
revolving,  402,  405 
Variations  in  discharge,   132,  307, 

320,  324 
Velocities  in   a   cross-section,   198, 

299,  308 

Velocity,  4,  18,  46,  48 
absolute,  69 
coefficient  of,  112 
critical,  262 
curves  of,  199,  308 
from  orifices,  48 
in  conduits,  268 
in  pipes,  218 
in  rivers,  308 
mean,  97,  199,  218,  267 
measurement  of,  96-103, 

200, 310 
of     approach,     56,     124, 

146,  168,  153 
of  stress  in  water,  12,  19, 
391 


Velocity  of  the  bore,  345 
of  waves,  494 
relative,  69 
to  move  materials,    288, 

324 

Velocity-head,  50,  76,  550 
Venturi,  J.  B.,  2,  60,  186 
Venturi  water  meter,  94,  198,  233 
Vermeule,  C.  C.,  349 
Vertical  jets,  49,  113,  192,  558 
orifices,  116-121,  139 
turbines,  445 
wheels,  423 

Vessel,  emptying  of,  59 
moving,  70 
revolving,  72 
Vortex  whirl,  60 

Waste  of  water,  244 

weirs,  157 
Water  barometer,  10,  19 

boiling  point  of,  10 

compressibility,  n 

dynamic  pressure,  375-411 

freezing  of,  6,  18 

hammer,  390 

mains,  185,  220,  246 

maximum  density,  8,  18 

measurement  of,  82,  131 

meters,  93 

physical  properties,  5 

pipes,  35,  44 

power,  355-374 

pressure  of,  22,  24,  31 

supply,  347-354,  528 

surface  of,  6,  24 

vapor,  498 

waste  of,  244 

weight  of,  7,  19,  474,  546 
Water-pressure    engine,    434,    465 
Water  wheels,  356,  362,  407,  412- 

.    436, 523 
Waterwitch,  483 
Watt,  J.,  510 
Waves,  373,  385,  492 
Webber,  C.  A.,  513,  516 
Weighing  water,  82,  358 


INDEX 


585 


Weight  of  ice,  6,  9,  19 
masonry,  42 
mercury,  10,  88 
sand,  27 
sewage,  9 

submerged  bodies,  26 

water,  7,  18,  474,  546 

Weirs, 

Weisbach,  J.,  2,  112,  113,  213,  260, 

378,  380,  445 
Wetted  perimeter,  265 
Wheel  pit,  466 
Wheels,  breast,  418,  434 

horizontal,  425,  443 
impulse,  423-432 
overshot,  415,  433 
reaction,  410,  437,  462 
turbine,  437-472 
undershot.  420,  434 


Wheels,  vertical,  423,  445 

Whirl  at  orifice,  60 

Whitehurst,  J.,  517 

Wide  crests,  156,  556 

Williams,  Hubbeil,  and  Fenkell,  91 , 

104,    201,   2J3 

Woltmann,  R. ,  99 
Wood  conduits,  276,  282 

pipes,  205,  254,  289 
Work,  defined,  4,  355 

in  propulsion,  479 

in  pumping,  496 

of  friction,  210,  269,  480 

of  motors,  413-470 

of  vanes,  399,  405 

units  of,  4,  1 8 


Young  man,  17,  506,  540 


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Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Lob's  Electrochemistry  of  Organic  Compounds.  (Lorenz.) 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments 8vo,  3  oo 

Low's  Technical  Method  of  Ore  Analysis 8vo,  3  oo 

Lunge's  Techno-chemical  Analysis.  (Cohn.) , i2mo,  i  oo 

Mandel's  Handbook  for  Bio-chemical  Laboratory I2mo,  i  50 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe .  .  i2mo,  60 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Matthew's  The  Textile  Fibres 8vo,  3  50 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  ,i2mo,  i  oo 

Miller's  Manual  of  Assaying. .  . i2mo,  i  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) ....  i2mo,  2  50 

Mixter's  Elementary  Text-book  of  Chemistry I2mo,  i  50 

Morgan's  Elements  of  Physical  Chemistry I2mo,  3  oo 

*  Physical  Chemistry  for  Electrical  Engineers izmo,  i  50 

4 


Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo,  5  oo 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,  2  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) I2mo,  i  50 

"               "           "             Part  Two."     (Turnbull.) i2mo,  200 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,  5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) 12010,  i  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  23 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's  Air,  Water,  and    Food  from  a  Sanitary  Stand- 
point. .  .  -. 8vo,  2  oo 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Cost  of  Food,  a  Study  in  Dietaries 12010,  i  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. 

Non-metallic  Elements.) 8vo,  morocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo,  i  oo 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo 

*  Whys  in  Pharmacy I2mo,  i  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) 8vo,  2  50 

Schimpf's  Text-book  of  Volumetric  Analysis I2mo,  2  50 

Essentials  of  Volumetric  Analysis I2mo,  i  25 

*  Qualitative  Chemical  Analysis 8vo,  i  25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  oo 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  TiUman's  Elementary  Lessons  in  Heat 8vo,  I  50 

*  Descriptive  General  Chemistry 8vo,  3  oo 

Treadwell's  Qualitative  Analysis.     (Hall.) 8vo,  3  oo 

Quantitative  Analysis.     (Hall.) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  i  50 

.  *  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Beet-sugar  Manufacture  and  Refining Small  8vo,  cloth,  4  oo 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  oo 

Wassermann's  Immune  Sera :  Haemolysins,  Cytotoxins,  and  Precipitins.    (Bol- 
duan.)   I2mo,  i  oo 

Well's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students I2mo,  i  50 

Text-book  of  Chemical  Arithmetic i2mo,  i  25 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes I2mo,  i  50 

Chlorination  Process i2mo,  i  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry I2mo,  2  oo 

5 


CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS   OF    ENGINEERING 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments lamo,  3  oo 

Bixby's  Graphical  Computing  Table Paper  19*  X  24*  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional.) 8vo,  3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage I2mo,  i  50 

Practical  Farm  Drainage I2mo,  i  oo 

*Fiebeger's  Treatise  on  Civil  Engineering 8vo,  5  oo 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  I/es's  Stereotomy ' 8vo,  2  50 

Goodhue's  Municipal  Improvements I2mo,  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Howe's  Retaining  Walls  for  Earth i2mo,  i  25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscoit  acd  Emory.) .  I2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.). .8vo,  5  oo 

*  Descriptive  Geometry 8vo,  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy .       Svo,  2  50 

Merriman  and  Brooks's  Handbook  for  Surveyors.  ... i6mo,  mor&v-  -  oo 

Nugent's  Plane  Surveying 8vo,  3  3~ 

Ogden's  Sewer  Design I2mo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

RideaPs  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i  23 

Wilson's  Topographic  Surveying 8vo,  3  50 


BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  oo 

*       Thames  River  Bridge 4to,  paper,  5  oo 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  50 


Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  . .  .8vo,  3  oo 

Design  and  Construction  of  Metallic  Bridges 8vo,  5  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II.  .. Small  4to,  10  oo 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Howe's  Treatise  on  Arches 8vo,  4  oo 

Design  of  Simple  Roof- trusses  in  Wood  and  Steel 8vo,  2  oo 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,    ro  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges : 

Part  I.     Stresses  in  Simple  Trusses 8vo,  2  50 

Part  II.     Graphic  Statics 8vo,  2  50 

Part  III.     Bridge  Design 8vo,  2  50 

Part  IV.     Higher  Structures 8vo,  2  50 

Morison's  Memphis  Bridge 4to,  10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,    2  oo 

Specifications  for  Steel  Bridges i2mo,    i  25 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume 8vo,    3  50 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo,  2  oo 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering. 8vo,  6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

Hydraulic  Motors 8vo,  2  oo 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Folwell's  Water-supply  Engineering 8vo,  4  oo 

Frizell's  Water-power 8vo,  5  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,-  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Bering  and  Trautwine.) 8vo,  4  oo 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  oo 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo,  4  oo 

Merriman's  Treatise  on  Hydraulics 8vo,  5  oo 

*  Michie's  Elements  of  Analytical  Mechanics .8vo,  4  oo 

Schuyler's  Reservoirs  for  Irrigation,  Water-power,  and  Domestic  Water- 
supply Large  8vo,  5  oo 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post.,  440.  additional. ).4to,  6  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams 4to,  5  oo 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 4to,  10  oo 

Williams  and  Hazen's  Hydraulic  Tables '. 8vo,  i  50 

Wilson's  Irrigation  Engineering Small  8vo,  4  oo 

Wolff's  Windmill  as  a  Prime  Mover , 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  oo 

7 


MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo,  5  oo 

Roads  and  Pavements.  .  .  . 8vo,  5  oo 

Black's  United  States  Public  Works Oblong  4to,  5  oo 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

Byrne's  Highway  Construction 8vo,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to,  7  50 

*Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  oo 

Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,  4  oo 

Rockwell's  Roads  and  Pavements  in  France 12010,  i  25 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines i2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Text-book  on  Roads  and  Pavements i2mo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Materials  of  Engineering.     3  Parts .  .8vo,  8  oo 

Parti.     Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

*    Constituents 8vo,  2  50 

Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  oo 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.).  . i6mo,  mor.,  2  oo 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel •  .8vo,  4  oo 


RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  I  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book. .  i6mo,  morocco,  5  oo 

8 


Dredge's  History  of  the  Pennsylvania  Railroad:    (1879) Paper,  5  oo 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills. 4to,  half  mor.,  25  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.  .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocce,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  oo 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,    5  oo 


DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "                                         "        Abridged  Ed 8vo,  i  50 

Coolidge's  Manual  of  Drawing 8vo,  paper  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Advanced  Mechanical  Drawing 8vo,  2  oo 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  H.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics ;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

MacLeod's  Descriptive  Geometry Small  8vo,  i  50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo,  i  50 

Industrial  Drawing.  (Thompson.) 8vo,  3  50 

Meyer's  Descriptive  Geometry 8vo,  2  oo 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  co 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.  (McMillan.) 8vo,  2  50 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo, 


Drafting  Instruments  and  Operations i2mo, 

Manual  of  Elementary  Projection  Drawing i2mo, 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo, 

Plane  Problems  in  Elementary  Geometry I2mo, 

9 


Warren's  Primary  Geometry I2mo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50 

General  Problems  of  Shades  and  Shadows 8vo,  3  oo 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo,  2  50 

Weisbach's    Kinematics  "and    Power    of    Transmission.        (Hermann    and 

Klein.) 8vo,  5  oa 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving 12 mo,  2  oo 

Wilson's  (H.  M.)  Topographic  Surveying 8vo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective 8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  i  oo 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  oo 

ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oa 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  12 mo,  i  oo 

Benjamin's  History  of  Electricity 8vo,  3  oo 

Voltaic  Cell 8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).Svo,  3  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  oo 
Dolezalek's    Theory   of    the    Lead   Accumulator    (Storage    Battery).      (Von 

Ende.) i2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power I2mo,  3  oo 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,  2  50 

Hanchett's  Alternating  Currents  Explained i2mo,  i  oo 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Holman's  Precision  of  Measurements 8vo,  2  oo 

Telescopic   Mirror-scale  Method,  Adjustments,  and   Tests.  .  .  .Large  8vo,  75 

Xinzbrunner's  Testing  of  Continuous-current  Machines 8vo,  2  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

Le  Chateliers  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo,  3  oo 

Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo,  3  oo 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  Svo,  each,  6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light Svo,  4  oo 

NiaudetVElementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo,  2  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .8vo,  i  50 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  I .8vo,  2  50 

Thurston's  Stationary  Steam-engines Svo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat Svo,  i   50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  Svo,  2  oo 

Ulke's  Modern  Electrolytic  Copper  Refining Svo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law Svo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States Svo,  7  oo 

*  Sheep,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence Svo ,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  Svo  5  oo 

Sheep,  5  50 

Law  of  Contracts Svo,  3  oo 

Winthrop's  Abridgment  of  Military  Law I2mos  2  So 

10 


MANUFACTURES. 

Bernadou's  Smokeless  Powder— Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

Holland's  Iron  Founder I2mo,  2  50 

"The  Iron  Founder,"  Supplement I2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Eff rent's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

^itzgerald's  Boston  Machinist I2mo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf's  Steel.     A  Manual  for  Steel-users 12 mo,  2  oo 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. 8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish; 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Spalding's  Hydraulic  Cement I2mo,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Beet-sugar  Manufacture  and  Refining Small  8vo,  4  oo 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book I2mo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,  4  oo 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,    i  50 

*  Bass's  Elements  of  Differential  Calculus izmo,    4  oo 

Briggs's  Elements  of  Plane  Analytic  Geometry lamo, 

Compton's  Manual  of  Logarithmic  Computations i2mo, 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo, 

*  Dickson's  College  Algebra Large  i2mo, 


*       Introduction  to  the  Theory  of  Algebraic  Equations Large  xarno, 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo, 

Halsted's  Elements  of  Geometry 8vo, 


Elementary  Synthetic  Geometry 8vo, 

Rational  Geometry i2ino, 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:   Vest-pocket  size. paper,  15 

100  copies  for  5  oo 

*  Mounted  on  heavy  cardboard,  8  X  10  inches,  25 

10  copies  for  2  oo 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  .Small  8vo,  3  oo 

Johnson's  CW.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  8vo,  i  50 

11 


Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,     i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,    3  50 
Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  lamo,     i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,    3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.).  i2mo,    2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,    3  oo 

Trigonometry  and  Tables  published  separately Each,    2  oo 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,     i  oo 

Mathematical  Monographs.     Edited  by  Mansfield  Merriman  and  Robert 

S.  Woodward.  .  .' Octavo,  each     i  oo 

No.  i.  History  of  Modern  Mathematics,  by  David  Eugene  Smith. 
No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
No.  3.  Determinants,  by  Laenas  Gifford  Weld.  No.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  No.  5.  Harmonic  Func- 
tions, by  William  E.  Byerly.  No.  6.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors, 
by  Robert  S.  Woodward.  No.  8.  Vector  Analysis  and  Quaternions, 
by  Alexander  Macfarlane.  No.  9.  Differential  Equations,  by 
William  Woolsey  Johnson.  No.  10.  The  Solutism  of  Equations, 
by] Mansfield  Merriman.  No.  u.  Functions  of  a  Complex  Variable, 
by  Thomas  S.  Fiske. 

Maurer's  Technical  Mechanics 8vo,    4  oo 

Merriman  and  Woodward's  Higher  Mathematics 8vo,    5  oo 

Merriman's  Method  of  Least  Squares 8vo,    2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,    3  oo 

Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,    2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,    2  oo 

Trigonometry:  Analytical,  Plane,  and  Spherical . i2mo,    i  oo 


MECHANICAL  ENGINEERING. 
MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "                  "                "        Abridged  Ed 8vo,  150 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  oo 

Carpenter's  Experimental  Engineering 8vo,  6  oo 

Heating  and  Ventilating  Buildings. 8vo,  4  oo 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  CoaL     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

.     Flather's  Dynamometers  and  the  Measurement  of  Power. i2mo,  3  oo 

i          Rope  Driving I2mo,  2  oo 

^    Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

12 


Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  I  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.    (Pope,  Haven,  and  Dean.) .  .  8vo,  4  oo 

MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

MacFar land's  Standard  Reduction  Factors  for  Gases 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richard's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

Thurston's   Treatise   on   Friction  and   Lost   Work   in   Machinery   and   Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics .  I2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  500 

Wolff 's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines, , 8vo,  2  50 


MATERIALS  OP  ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

Strength  of  Materials I2mo,  i  oo 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines I2mo,  i  oo 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their      . 

Constituents 8vo,  2  5^ 

Text-book  of  the  Materials  of  Construction 8vo,  5  oo 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

13 


Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

SteeL 8vo,  4  oo 

STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram i2mo,  i  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .      i6mo,  mor.,  5  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  I  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  oo 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  oo 

Heat  and  Heat-engines 8vo,  5  oo 

Kent's  Steam  boiler  Economy 8vo,  4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  T  50 

MacCord's  Slide-valves 8vo,  2  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator I2mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors   8vo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  oo 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo,  4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) I2mo,  i  25 

Reagan's  Locomotives:  Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  oo 

Sinclair's  Locomotive  Engine  Running  and  Management I2mo,  2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice I2mo,  2  50 

Snow's  Steam-boiler  Practice 8vo,  3  oo 

Spangler's  Valve-gears 8vo,  2  50 

Notes  on  Thermodynamics i2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Handy  Tables 8vo,  i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory 8vo,  6  oo 

Part  II.     Design,  Construction,  and  Operation 8vo,  6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  oo 

Stationary  Steam-engines 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice I2mo,  i  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation 8vo,  5  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  oo 

Whitham's  Steam-engine  Design 8vo,  5  oo 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  ..8vo,  4  oo 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures   8vo,  7  50 

Chase's  The  Art  of  Pattern-making I2mo,  2  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Notes  and  Examples  in  Mechanics '. 8vo,  2  oo 

Compton's  First  Lessons  in  Metal-working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe I2mo.  i  sa 

14 


Cromwell's  Treatise  on  Toothed  Gearing I2mo,  I  50 

Treatise  on  Belts  and  Pulleys i2mo,  ~  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .i2mo,  i  50 

Dingey's  Machinery  Pattern  Making izmo,  2  oo 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 410  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  400 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  IL Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Hall's  Car  Lubrication i2mo,  i  oo 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  8vo,  2  oo 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods '. 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.). 8vo,  4  oo 
MacCord'*  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

Maurer's  Technical  Mechanics.  .  .  .• 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Elements  of  Mechanics 12010,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric.  . I2mo,  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  co 

Sinclair's  Locomotive-engine  Running  and  Management I2mo,  2  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines 12 mo,  i  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design , .  m 8vo,  3  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

I2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   (Herrmann — Klein.).  8vo,  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein. ).8vo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics. 8vo,  3  oo 

Principles  of  Elementary  Mechanics i2mo,  i  25 

Turbines 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

15 


METALLURGY. 


Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  50 

Vol.  II.     Gold  and  Mercury 8vo,  7  50 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional.) i2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess. )i2mo.  3  oo 

Metcalf' s  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.). . .  .  i2mo,  2  50 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

Smith's  Materials  of  Machines I2mo,  i  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  oo 

Part    II.     Iron  and  Steel • 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 


MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  oo 

Map  of  Southwest  Virignia Pocket-book  form.  2  oo 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy f.  . .  .Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  8vo,  i  oo 

Text-book  of  Mineralogy 8vo,  4  oo 

Minerals  and  How  to  Study  Them i2mo,  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  oo 

Manual  of  Mineralogy  and  Petrography lamo,  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects I2mo,  oo 

Eakle's  Mineral  Tables 8vo,  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.). Small 8vo,  2  oo 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses 8vo,  4  oo 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 
Rosenbusch's   Microscopical  Physiography   of   the   Rock-making  Minerals. 

(Iddings.).  .  .  • 8vo,  5  oo 

*  Tollman's  Text-book  of  Important  Minerals  and  Rocks 8vo,  2  oo 


MINING. 

Beard's  Ventilation  of  Mines i2mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia. 8vo,  3  oo 

Map  of  Southwest  Virginia Pocket-book  form  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  oo 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4to,hf.  mor.,  25  oo 

Eissler's  Modern  High  Explosives 8vo>  4  oo 

16 


Fowler's  Sewage  Works  Analyses i2mo,  2  oo 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States i2mo,  2  50 

Ihlseng's  Manual  of  Mining 8vo,  5  oo 

**  lles's  Lead-smelting.     (Postage  pc.  additional.) i2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

O'Driscoll's  Notes  on  the  'treatment  of  Gold  Ores 8vo,  2  oo 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process ' i2mo,  •  i  50 

Hydraulic  and  Placer  Mining I2mo,  2  oo 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo,  i  25 


SANITARY  SCIENCE. 

Bashore's  Sanitation  of  a  Country  House I2mo,  i  oo 

FolwelTs  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oo 

Water-supply  Engineering 8vo,  4  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works I2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Ogden's  Sewer  Design i2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  25 

*  Price's  Handbook  on  Sanitation I2mo,  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  oo 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  oo 

Richards  and  Woodman^  Air.  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies ".  .8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) I2mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 


MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff  and  Collins.).  . .  .Large  I2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo,  4  oo 

Haines's  American  Railway  Management I2mo,  2  50 

Mott's  Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  oo 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1 894.. Small  8vo,  3  oo 

Rostoski's.Serum  Diagnosis.     (Bolduan.) I2mo,  i  oo 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  oo 

17 


Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Winslow's  Elements  of  Applied  Microscopy i2mo,  i  50 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture:  Plans  for  Small  Hospital.  1 2mo,  i  25 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar I2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  oo 

Letteris's  Hebrew  Bible. 8vq,  2  25 

18 


YC  3 


256302 


1903 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


